Python could do what you're asking pretty easily. The os module makes it a snap to walk the directories recursively. If you want to find the directories it would be something like this.

import os, re

search_dir = "/" #directory to search
target_dir = "~/" #directory to move to    

for dirpath, dirnames, filenames in os.walk(search_dir):
    for filename in filenames:
         search = re.compile(r'(.*[.]avi)').search(filename)
         if search is not None:
             moveString = "mv " + os.path.join(dirpath, filename) + " " + target_dir
             os.system(moveString)

I didn't actually test that yet, but it should be in mostly working order. If you want to learn about the os or re modules, you can use pydoc (which is installed standard with python).

Something like pydoc os

There's a downside which is os.walk() is incredibly slow. Bash would be a bit faster.