Rescue Mission Ranks also go up as you go up in Rescue Team Ranks

Unlockable - How to Unlock

Bronze - Acquire 50 Rescue Points
Diamond - Acquire 7500 Rescue Points
Gold - Acquire 1500 Rescue Points
Lucario - Acquire 15000 Rescue Points
Platnium - Acquire 3000 Rescue Points
Silver - Acquire 500 Rescue Points