Currently, in the chat-rooms the avatar is shown from the parent site that a user selected in the chat profile. So, throughout SE network in (any) chat-room (talking about main chat server i.e not MSE or SO), the same avatar is shown.
But as we know, sometimes users don't use the same avatar on all communities, but use different avatars according to the site. So, I think when a user enters the chat-room, the avatar from the site related to that chat-room should be shown.
As the chat-room is nicely related to specific site and users want the avatar according to the specific site, chat-room should use the avatar of related site of chat-room instead of using the same avatar of a parent site in all communities.
Note: In case the user doesn't have account on the site related to chat-room, use avatar of parent site.