The violin bowing technique for string crossings is different for different tempi

Violin string crossings with the shoulder

For slower string crossings that don’t go back and forth, use the shoulder and move the whole arm. This is the first string crossings technique you’ll learn on the violin.

Violin string crossings with the elbow

For slightly fast string crossings, you can move just the lower arm.

Violin string crossings with the wrist

For smooth string crossings in a higher tempo, usually back and forth, you can cross string just with the wrist.

Violin string crossings with the fingers

Although I recommend using finger action in all type of string crossings to make them smooth, use only the fingers when you have to cross strings back and forth in a high tempo.

It’s possible to use combinations of these techniques!

Certainly in complex string crossings over three or four strings, a combination of shoulder, elbow, wrist and fingers will make your string crossings smooth and clean.