Hi,
As long as the LeafBridge triggers the save_post hook, you don’t need to do anything. That hook is triggered whenever a new post or post object is created/changed and saved to the database. Usually properly coded migration/sync plugins trigger this hook for every single item they synced so the appropriate functions attached to that action can execute.
Ajax Search Pro listens to that action and whenever it’s triggered it updates it’s database with the new post data automatically.