Existing TCP streams pinned to old instance after moving Elastic IP to new instance

Yes. In the docs:

You can disassociate an Elastic IP address from a resource, and then associate it with a different resource. 
To avoid unexpected behavior, ensure that all active connections to the resource named in the 
existing association are closed before you make the change. After you have associated your 
Elastic IP address to a different resource, you can reopen your connections to the newly 
associated resource.