@grasstreeleaf
I can't tell you because I'm not a MIL. However, I can say how I saw my own mother reacted becoming a MIL. She was beside her self with excitement over an new grandchild but somewhat on that back foot because my SIL isn't her daughter and couldn't just push her way in without treading on toes. Although I think she did but became incredibly defensive at the same time. Thankfully things settled down but I did see tears...I was surprised because I couldn't believe she was so adamant on pushing her way in. She had been very easy going. But again she was incredibly excited at the new grandchild. I think people genuinely do find it difficult to negotiate and are unprepared for the emotions they feel.
I agree with this.
My own MIL is generally nice. We've always got on just fine. Close but not super close, not like mother and daughter but happy to spend time together. She's a normal, reasonable and quite thoughtful person!
When my eldest was born, she changed overnight and it really shocked me. She was demanding and selfish. I had a terrible time giving birth, literally baby and I were lucky to both survive it. Yet there she was demanding to visit the next day (and sulking because she couldn't as I was still in a high dependency unit not the ward and wasn't allowed any visitors except DH!). Once I moved to my room on the ward, the only person I wanted to visit was my mum and it was nothing to do with showing off the baby it was because I was really poorly and wanted my mum. Again, once my mum had visited she was on the phone to DH badgering to be allowed up on the basis that it 'wasn't fair'. My mum had helped me shower whilst DH sat with the baby and had a kip in the chair. She'd sat with me while I cried through trying to breastfeed topless in an effort to do the skin to skin thing. I looked a fright, I stank when she arrived because I'd not showered for 4 days and had only had a quick bed wash and I felt dreadful. I didn't want anyone else there not even my Dad or my siblings.
She came up a couple of days later (whilst I was still in hospital) and she was ok but barely asked how I was feeling was just all about taking photos of her and the baby. She then sulked again when I said no photos on social media please, we're not doing that at all.
Her behaviour continued - when baby was about 3 weeks old and we were home she had a huge strop because we'd arranged for her to come round for a visit and after a night of no sleep I'd managed to actually out the sleeping baby down in her moses basket and had dozed off in the sofa next to her. DH rang her and said I was exhausted and had fallen asleep so he was leaving me that way and we'd reschedule the visit. She said no, she'd just sit quietly and wait until we woke up! Er what, and watch me sleep?? I was literally in a skanky vest and pair of knickers on my sofa 
DH had to really put his foot down, she had a tantrum and in the end DH spoke to his Dad and got him to have a word. I woke up to a text message about how I'd ruined her day by cancelling.
She did calm down and also I got much firmer and assertive with her. We had a huge row in the end and she came to me a few days later and apologised. She said she was just over excited and loved the baby so much. I do genuinely believe that, she's not a horrible person at all but she just got a bit blinded and acted like a total arse.
Second baby she didn't behave like that (learned her lesson maybe and knew by then I'd not tolerate it!) and she is actually the most wonderful grandmother who dotes on my children. They have a fabulous relationship and we have a normal one like we did pre children.
My own mum was fabulous when my children were born... but I also suspect she could be a pain in the bum MIL when my DB and SIL have children. She's quite forthright and bossy generally which is fine for me because I'm used to it and just tell her to pack it in with me. But I will be keeping an eye out for it with my SIL and making sure she doesn't tread on toes there!