Before migrating user avatars to bbPress forum board, you are to have BuddyPress plugin installed and activated. This plugin expands user functionality by giving more handful features, one of which is custom avatars improvement.
Also, make sure that writable permissions (they are 777 or 755 depending on your server configuration) are set for default avatar folder on your target website. By default, for user avatars, the folder is as follows – /wp-content/uploads/avatars