Hi Kate,
That is most definitely something we want to do. One of our goals for Event Aggregator is to make it a true syncing resource between two sites that are both using The Events Calendar, Event Tickets and our other plugins in order to port all data from one site to another.
I found the ticket we have on file for this feature in our system and have added your vote to it so it does not go unnoticed. It will also allow us to follow-up with you directly once it is completed, though I do not want to get your hopes up too quickly because the feature has yet to be earmarked for a specific release on our roadmap.
Does this help answer your question? Please let me know.
Cheers,
Geoff