It's helpful to spend more time on your wrist stretches at the beginning of your training session. You can use our wrist article, which has information and advice on improving and maintaining wrist health.
When it comes to the locomotive movements themselves, the bear, monkey, and frogger, something that can help is rotating your hand outwards when it's placed on the ground, instead of straight ahead. You can also try having a small folded hand towel under the heel of your palm so that the wrist doesn't have to extend as much. So you grip it and have it move as you move.
For more info and fun variations on locomotion, check out these articles: