I don't understand why he doesn't have contingency plans? The simple fact of the matter is that his ex should be able to rely on her children's dad to have their children when he says he will. She should be able to plan her life around that - whether that be work, chilling time, going out, or cleaning the house. She should also be able to plan ahead - book tickets to a concert or a hotel overnight, for example.
What you have here is a man who is either going to get his partner to sort out the issue, or have his ex wife step up and deal with it. He is basically using the women in his life as his back up childcare.
I have been a RP for some 12 years and I have lost count of the emergencies I have had to deal with, the times I've needed to be my own back up, not to mention my being unwell, unable to drive through injury or other 'stuff' that seems to happen. Not once has my ex stepped in. I have dealt with it all - often at great cost (taxis, for example) and somehow managed to keep my job and keep my children alive and well. So if I can do it, why can't he? Why is recognising that sometimes, things are outside of your control and that you're going to have to think outside the bloody box a bit to deal with it so hard for so many?
What will actually happen is the ex will miss out on whatever it was she had planned. Probably at some cost to herself - financially of course, but also possibly emotionally/mentally as well.
It's not a common courtesy to travel 3 hours with your small child to pick up the children of someone you "barely know" because their Mum has got plans that night
The OP is sleeping with the children's father? She knows him. The children are not just their mother's. They have a father who should be arranging to collect them - somehow - on his contact time. Any favour being done here is a favour to the OP's partner and children, not the ex.