Band dislocations are a great way to mobilize the shoulder because they take your shoulder through a full range of motion. This warm-up also activates the rotator cuff and upper back muscles.
Arm circles warm up the shoulders and chest. Make big circles as wide as you can, while keeping your elbows straight.
Be sure to go forward and backward.
Next, we need to practice extending the shoulder joint. The reverse bridge does a great job at improving shoulder extension range of motion, while also activating the muscles of the upper back.
To do calisthenics effectively, you need healthy wrists. This is especially important for push-up and handstand exercises.
The best way to mobilize the wrist is wrist circle variations.