Are you talking about getting yourself to work for taking the kids out?

You are expected to get yourself to work and back on your own dime for any job.

If you take the kids out on public transportation for the benefit of the kids (field trip, etc), then 100% should be paid by employer.

Most of my families that required me to take kids anywhere on the train paid for my monthly unlimited card ($127/month in NYC now). A one kept a loaded metrocard in a drawer for me separate from my personal card to use when I took the kids anywhere but I paid for my own commute. I typically ended up not using it anyway, because I had a monthly unlimited and the kids were small enough to ride free, and they appreciated saving the $20 or so a week. It depends on a lot of variables I think, so what you want to may differ if the kids go a lot of places or you strictly want to pay her commute.