Another thing we did at our last company when we consolidated databases on SQL instances is that we (I was part of the networking team) also basically created a DNS CName record for each application's DB.

So App1 had a CName record that pointed to ServerName1 AName record. So if we found a particular DB was causing too much stress on ServerName1, we could move or restore it on ServerName3 and then just edit the App1 CName record to point to the new server. It was an evening or weekend job, but it allowed us to make changes/moves that were invisible to the end user for all intents and purposes.