I agree that communication is the key. It sounds like there is something else going on.
Many women would never ask their Mother-in-laws for something they would ask of their mums. Perhaps she is struggling but does not want to admit it. That is completely normal. Why not suggest you MIL comes to visit regularly rather than her moving back home?
But I have to say her home is with you not with Mum & Dad, you need to find out why she is so keen to move away. Hopefully it is not as serious as she might think it is (for example a bit of Post Natal Depression) but you also need to be prepared that it might be something more serious.
Either way you need to find out. Best of luck!