The problem here is you have described two very different situations in your posts. First situation, Auntie is babysitting, which FYI she doesn't actually do very often, rubbish Aunty, and not only that but she refuses to come by train to pick them up! She must have an empty life without children, so why can't she run around after mine?
Second situation. Aunty has asked to have the kids for the weekend. We are really pushed for time and it isn't really convenient this weekend as we are helping out my ill father. I asked Aunty if she was able to come by train to collect the kids and she replied "no". It's going to be really difficult to get the kids to her, I'm not sure how we will manage it.
First situation you are being unreasonable, second one I can see how that is frustrating and annoying.
My kids used to go to MILs for tea on a Thursday after school, initially because I was working, but then I changed my working pattern and there was no need, in fact I didn't like them going as I like having them at home after school, I miss them all day! But I thought it was nice for them to see their grandmother and grandfather. But then it got to a point where all she did was complain about them, how difficult it was having them, how tired she was, how badly behaved they are etc etc and for us getting them home was really inconvenient as FIL wouldn't drive them (a 30 min walk at the end of a long day, often in pouring rain it seemed!!) so we stopped them going. She rarely bothers with them now. When I was working, we would have put up with it all because they were doing us a favour, free childcare. Once I wasn't working and it was, I thought, more for MIl and FIL (kids weren't fussed), it became a different situation altogether to put up with their moaning etc!
In your situation though OP I think you need Aunty to have the kids this weekend so you are free to help your father, so she is doing you a favour., and your OP is probably closest to the real scenario. So I think YABU, although her short text is a bit rude, that would annoy me too.