It depends on the linear pattern played. This image covers the basics:

Source: https://imslp.org/wiki/Mastering_the_Scales_and_Arpeggios_(Cooke%2C_James_Francis)
The basic difference is how you change position.
The first group of examples uses linear patterns that stay within the range of one octave and then shift position simply by moving the arm and hand up or down the keyboard. In those cases it's OK to use the thumb on black keys.
The second group labeled with Roman numerals uses linear patterns exceeding one octave and passing the thumb under to change position. When passing the thumb under for those you don't want to pass the thumb under to a black key. You especially don't want to go from a white key to passing the thumb to a black they. It's just a matter of the shape of the hand and the keyboard. Thumb on a white key and fingers on the black keys fits the hand.
Notice that the fingering for...

...is essentially the same as...

...but the F
natural where the thumb is used falls on different metrical positions. One is one the beat, the other is off the beat.
You can see that same kind of pattern for the other chords rooted on a black keys.
You could think of it as just one set of fingerings for the three chord positions (inversions) but various linear patterns require starting with differing fingers and changing position at different times.