There are several movements required for an arpeggio and although having only a few of them is okay and acceptable, you should have them all. Also, if you have just one of the several improper movements, they can hinder the proper movements. You can have the best shoes in the world but if there is a pebble in there . . . . . So, It would behoove you not to try anything I suggest but instead find a good teacher who knows physics, anatomy and ergonomics and work on all these movements individually before putting them all together for, YOU CAN INJURE YOURSELF. I am only going to talk about a CEG arpeggio as every key is different.
Your fingers are all different lengths and many pianists attempt to curl and equalize them but that just creates tension and eventual injury, not to mention uneven playing. To accommodate those different lengths you need to have in/out motions to make them FEEL equal. Because the keys are a fulcrum, they are lightest on the outside edges and in/out will facilitate playing out there. If you have ever experimented with playing on a see saw, you know you feel heaviest the further out you sit. That is the secret to controlling the weight of your opposing partner. If you play your thumb down on an outside edge that places your unused fingers in the black key area where the keys are heaviest. When you play your two, you move out to the edge and that takes your thumb off the key edge entirely. Your three finger takes all the fingers off. Then you move in with your four and further in with your five. That is in/out and enables you to play on the outside edge where the keys are lightest.
There are a few ways to play down. The thumb's abductor is its weakest muscle and that is what most pianists use to play down. You can however use your pronator which is near your elbow and play the thumb down without using your thumb's abductor at all (rotation like if you tremolo). You can also let the weight of the arm play the thumb down. Play a chord and notice you play with the weight of the arm and don't use any finger. You can do that with the thumb and all the other fingers, too. To avoid falling off the keys, it helps to have a forward shift into the key. THIS IS NOT THE SAME AS PRESSING INTO THE KEYBED. As you come out you are also shifting in. This comes from the elbow, shoulder and forearm. When you combine in/out, gravity, forward shift, pronation and supination you don't use any finger muscle. Actually, your fingers have no muscle but are moved by the muscles of the forearm. This combination of movements make the fingers feel and play effortlessly. When you use your fingers flexors exclusively, you will develop problems. Your fingers are designed for poking and gripping, not playing down. So up and down or gravity is your best friend at the keyboard. But, NEVER press into the keybed. Instead try to play to the point of sound. Slowly press a key down without making a sound. Feel that little bump? Press beyond it into the keybed. That bump is the point of sound. Also, if you press down you can't lift up. Your arm and finger can only go in one direction at a time. If you play to the point of sound and allow a follow through, your arm can set up the next note. Sort of like throwing a ball or punching someone in the face. After you release or make impact, there is a follow through. If there is no follow through you will hit the keybed and create tension. Like hitting a brick wall. Don't ever challenge the piano to a duel of strength, it will win. Instead, develop the skill of playing to the point of sound, also known as carreszando.
A movement which gets in the way of many pianists is abduction. When you abduct your fingers (spread them out), two things happen, your hand gets pulled in two directions simultaneously and this creates tension and disrupts the direction the arm is trying to move in. If you have ever run in a three legged race, you know both you and your partner must move in the same direction at the same time. If one of you is faster than the other, anarchy reigns. The same is true for your five fingers. Secondly, it pits two muscles against one another, your abductors and flexors. With all five fingers together, wave bye bye. Now abduct them and wave bye bye. Feel the tension? Why would you play like this? Keep your fingers together at all times.
So if you don't abduct, how do you cross the distance required for arpeggios? Rotation of the pronator and supinator from the elbow FROM THE ELBOW FROM THE ELBOW FROM THE ELBOWhelps as does leading with the arm in the direction of the pronator. If you fake rotate from the wrist you will develop problems. The elbow will propel the arm and place the fingers where they need to be. By doing this there is never a reason to cross your thumb under because the supination and moving the arm up will place the thumb on the next key and you simply need to pronate down to it. Also, when you cross the thumb under the palm, its tendon intersects with your index finger's long flexor tendon and they will grind together leading to future thumb injuries.
Two movements can get in the way of this: unlar and radial deviation. That is twisting the wrist left or right. Many pianists will twist the wrist in the direction they are going but this not only creates tension and can lead to ganglion cysts caused by cracks in the tendon sheath and leaks of synovial fluid where it should not be, but you then have to take the time to undo the twist which disrupts the flow. "Strength" of the fingers comes from having the forearm aligned directly behind the finger being used. Some pianists think their four and five fingers are weak but they are not if the arm is aligned behind them. A twist will break the alignment of the arm behind the finger. There are two elbow movements to rectify this which I don't have a name for. It is the raising or lowering of the elbow in combination with pronation and supination. I suggest watching pianists such as Valentina Listisa who often plays sleeveless but don't watch her hands, watch her elbows. Adam Makowicz often exaggerates this movement, too. He will exaggerate it once then suddenly minimize it so you have to catch it the first time. Again, watch the elbow, never the fingers. Fingers are an extension of the arm and watching them is fruitless. You'll see the result and not the action.
Another movement which will help is leaning in the direction of the arpeggio and leaning forward in order to get the hand where you want it to be. Think of this like casting a fishing pole. The big movements make the little movements more effortless. If you cast from just your fingers, you will fail. If you cast from the wrist, it is better but still bad. Cast from the elbow even better. Applying the shoulder would help a lot. But, use your whole body and you will procure maximum control and power over the pole. Too often pianists sit at the piano and try to play only from the fingers, they static load and injure themselves. Just a little aid from the big muscles makes the work of the little muscles effortless. Remember from HS Physics class, every motion must have an equal and opposite motion. There is no down without first and up. Again, play a chord. What is the first thing your arm does? It raises up. Your fingers need an up from somewhere, too. Hanon suggested raising a finger but this just isolates it and creates pulls and tensions.
So to play this CEG arp: The thumb plays down/in/pronated on the edge of the C. Next the index finger plays the E by the arm moving over, a little supination, a little gravity, a little out, a little forward but straight down. The middle finger plays the G with all the same movements but you add a little extra supination which raises the thumb up as the arm continues moving up the keyboard. The thumb then pronates down to the octave C because it is now right over it. Etcetera. Down is the opposite except you use that elbow trick when you play the C to place the middle finger over the G and straight down as your arm descends and you subtly supinate so you don't have to cross the thumb under for descending.
Special care must be taken when playing black keys because they are higher. This requires a different sort of up. As I said, each finger must come straight down on each key. If there is the slightest lean your weight can throw the balance of the arm off. This is like walking up stairs, Your ascending foot must rise higher than the next step in order to come straight down on it. If you are too low you will trip up stairs. The same thing can happen with our playing. All these movements must come together so that each finger has interdependence yet unified with the arm moving in one direction. All the up/down/in/out/rotations and single directions make all the fingers equal, powerful and gives them the illusion of independence. This is required whenever playing a black key.
Ultimately you should work on all these movements one at a time with a teacher. Your brain is a super computer which relies on what we call muscle memory and if you developed an improper movement from the very first day you touched a piano, that improper movement is hardwired into your brain and you must rewire it. The problem is the improper movement is there forever and you must be cognizant of it and never allow it to resurface. Should you ever practice while your body is cold or you are nervous, the improper movement will take over and put you at great risk for injury. We have all floundered in performance only to say "I played it perfectly at home." That is because while in performance, muscle memory reared its ugly head because we let our guard down. This is also why our first teacher should always be the best. The goal of a student should not be to learn to play quickly but to learn to play properly. Bad habits are forever but can be overwritten.
Should you ever feel rusty after missing a few days of practice, it is because you are doing something wrong. Your brain is remembering improper movement. If you wake in the morning and your fingers feel stiff, you are doing something wrong. That is from inflamed tendons from improper movement. Your tendons glide through a sheath which is lubricated with synovium. If you abuse your tendons they will feel stiff until they are relubricated with a little movement. Your fascial tissue could also be scarred which causes tightness. Also, when your tendons become inflamed, they can press on your median nerve causing median nerve entrapment also known by lay people as carpal tunnel syndrome. The cure is simple: Proper movement.
And another thing, never stretch. When you stretch you create tears in your muscle which causes the body to flood the site with warm blood to begin repairs. This gives us the illusion of warming up. If you stretch beyond the muscle you stretch tendons and can create micro tears but tendons don't have a blood supply so they don't heal. Instead your body places scar tissue there. Scar tissue doesn't stretch and will make you feel tight. The next time you over stretch you will tear the scar tissue which will create larger scar tissue patches resulting in inflamed tendons which will press on your median nerve. Again, this is why you feel rusty. Your brain knows the exact movement and measurement to strike a key from hardwiring but if the tendons are tight they will slip up the brain and your technique. What do we do? We use brute force to overcompensate. A downward spiral ensues. The good news is this will probably take years to cause a major problem. The bad news - it will happen.
I'm sure I am leaving something out. I ain't not no going to proof that. Which is all the more reason why you need to work with a knowledgeable teacher. Good luck finding one who knows all that. As I said, music is not just music, it is physics, anatomy and ergonomics.
Go with gravity!