As best I can guess, you are describing angioedema. Hives are swellings in the skin that look red, angioedema refers to swelling under the skin. Often, they go together. As you describe, they move around rapidly. Most of the time, they do that for a while and disappear. The vast majority of such cases have no cause, despite their reputation for being allergic.
For more specific information related to you, you'll need to see your doctor in person.
Best.
Dr. Rockoff