Like if you want to go from EADGBE to DADGBE, can you just tune the 6th string down and be done with it, or will that offset the bridge and throw the other strings out of whack?
Sadly, no. With almost any vibrato bridge, when you change one string, it will change the position of the bridge, and you’ll have to retune all the strings several times to find the bridge’s new home position.
That’s actually true for all string instruments, even guitars without vibrato bridges, it’s just much more obvious with a vibrato bridge. That’s because other things besides the bridge can move or change shape, primarily the neck. On acoustic instruments, the top often rises or falls slightly when tuning.
If you want to make sure you are tuned as well as possible, you should always go back and check each string until they are already in tune when you check them.
This actually is unrelated to whether the bridge is Floyd Rose or a non-locking model, the important distinction is whether it is set up as floating. Floating means the springs aren't so tight that the bridge is pressed against the top, but instead is balanced in an equilibrium between string- and spring tension. It it easy to find out whether a bridge is floating by pulling the lever, or simply pressing your hand flat on the bridge: if it's floating, the pitch will go up a bit, otherwise it will stay constant.
If the bridge is floating then yes, re-tuning any string will detune all the others, because the change in total string tension will shift the equilibrium. You could in principle re-tune the one string and then re-adjust the springs to compensate, but this isn't actually very practical – usually, instead the other strings will be tuned.
You can make a floating vibrato fixed by tightening the springs, or adding one more spring. The reason some guitarists prefer floating, apart from the ability to whammy up a bit, is that it allows a more gradual response. If it's fixed, you need to apply some initial force before the pitch changes at all, and then it goes down suddenly. But if you're only using the vibrato system for some violent drops anyway, then just don't set it up floating and you also won't have cross-string detuning problems (at least not more than with a fixed bridge). Again, all of this applies just as much to Fender- Ibanez- and Floyd Rose style vibrato systems. About Bigsby I don't know.