It's a matter of alignment of incentives. Within the company Department X knows that the company's success is their success, one team, one dream. On the outside their success comes from extracting as much as they can from the host entity, without killing it. You can't fault people for acting as they are incentivised to do of course. You can only set up a system of incentives to shepherd people towards the outcomes you want.

Emergencies for the business are not emergencies for the outsourced department.

Well of course not. If someone wanted to work for company Y, it's because, for whatever reason, they wanted to work for Y. If Y stabs them in the back, TUPEs them over to an outsourcing company, why would they go the extra mile? Good people whose goodwill has been wantonly squandered, and who know where all the skeletons are buried...