Yes, the service follows your partner wherever they go. The forwarding number InmateAid provides routes calls to your phone regardless of which facility your partner is calling from.
The one thing to be aware of is that different facilities use different phone carriers, and the carrier determines the rate structure. A number that produces the lowest rate at a county jail may not be the optimal number for the prison he moves to. When the transfer happens, let InmateAid know the new facility name by emailing aid@inmateaid.com and they will verify whether the current number is still giving you the best rate. If a different number would save you more at the new location, they will swap it out at no charge.
The service itself continues without interruption. Just make sure your partner adds the same forwarding number to his approved call list at the new facility once he has phone access there.