I'm probably being unreasonable. Just after some opinions really and how you would go about this.. I'll try not to ramble but there's a lot to this 
MIL got remarried shortly after DD was born. For two years all seemed fine on the surface but one day he up and left MIL and moved in with his boss and we found out he'd had a series of affairs etc. He worked away so we only saw him once a week, if that, but for two years he was "Grandad" to my DD.
Now ten months on, MIL is in the process of getting a divorce and has been dating a new man for the last three months, he also works away so if we wanted to get to know him, it would only be every other week really.
At the start of the relationship, MIL told DH and SIL they didn't have to meet him if they didn't want to. The relationship has moved very quickly and on several occasions she seems to have tried to artificially set up a meeting such as asking DH to pop over to fix something, while BF happened to be there. He is now also invited to Christmas.
We now have another baby and SIL has also had her first baby recently, we are all under the same mind set that we do not want our children having another part time "grandad". We are all happy for MIL if this man makes her happy, DH says he seems nice, but are we unreasonable for not wanting him involved in our kids lives? Especially as this is another part time relationship.
I think what is bothering DH and SIL most of all is after separating from their father, MIL never introduced any men she dated to them until they were much older and the relationship was serious. She isn't treating her grandchildren in the same manner despite wanting to be all about her grandkids since the separation.
Additionally, since this new relationship started, she has seen DD and DS twice, despite living a few minutes away. DD has stopped asking for her, she used to sleep over every Friday. She also told SIL she should be grateful she travels (90 minutes) to see her and her baby once a month, when in reality it is because her new BF lives in the next town and she is really just visiting him.