One of the most difficult things we can do on stringed instruments is play chromatics between strings because of the extended fingerings that we need to utilise to do so.
This short document demonstrates an exercise I developed for my students to build string-to-string confidence and allow them to play chromatic phrases in any position with full independence.
The diagram uses colours to signify which of the two adjacent strings is being played throughout the phrase. It also includes the fretting hand fingering to demonstrate where the shifts take place within the exercise.
It includes a 4-bar ascending sequence and a 4-bar descending sequence, which is when the fingerings repeat themselves between the next two adjacent strings.
This should be taken through as many keys as possible and if you can think of which notes you are playing as you go, rather than relying on the finger pattern, it can make it easier to play. If you struggle to do this because you don't know the notes on the neck well enough then this can also help you to develop better knowledge of the notes on the neck, and how they relate chromatically.
Corona Lesson - Chromatics Between Strings
Pdf file which includes two 4-bar ascending and descending exercises.