Most Helpful Girl
Offering is always nice, but it's reasonable to each pay your own way each time you visit.
When I had an LDR, we'd always take turns flying back and forth, and we each paid our own way.
I wouldn't start mixing money unless you are really serious about her; serious enough to marry her. Otherwise, it's just an expensive date and you'll be throwing your money away on someone when you could be putting it towards a different girl.