Use indexed shifters on new bike
Shifters don't care about wheel/rim size.  They only care about how many gears there are to shift across, and once you hit 10~11 speed then it also matters whether the bike is using a road or MTB groupset.
Answer Yes you can move shifters to another bike, as long as the speeds match, and the handlebar diameter is the same.
Your bike is a 3x on the front and a 8x on the rear, so the new bike would have to be the same.
If you have integrated shifters (where the brake lever and the shifter are in the same unit) then the brakes will have to match too - you couldn't use brake levers from V-Brakes with disk brakes.

 There are some situations where you could get away with a mismatch, but they'd be unusual, like using your 8 speed rear shifter on a 6 or 7 speed rear block, and having one or two dead positions.  Not recommended.