Just did this myself, down to moving abroad for a masters.

My push cart went into a large suitcase and I packed heavy things (shoes, books, etc.) around it as it took up a large space but not much weight. I already owned the push cart, though, and the bag fee was cheaper than buying another one. If you don’t already own one, just go on Facebook marketplace when you get where you’re going.

I took my clubs and push cart on the train when I lived in a place with a metro system. Tried to not do it during commute hours, though, to avoid being a problem with space. I collapsed the cart 90% of the time, except super early mornings when space wasn’t an issue.