As a practical matter, it will probably be too much hassle to get signed up with new flight schools in each city/country you visit for only one or two lessons, especially in the US where you need certain classes of visa (no VWP), permission from DHS, etc.
Also, FAA and ICAO/EASA rules can be pretty different. If you know one well, you can fairly easily learn the other (or just muddle along without anyone noticing), but I wouldn't want to mix the two during primary training. Learning one set of rules is hard enough.
The only case where this might make sense is if you'll be visiting the same city repeatedly and can get the same instructor every time, which eliminates most of the overhead. I did this (both cities in US), and my home instructor would give me a specific list of things to work on for each trip, and when I came back, I'd show what I'd learned and we'd move on to the next thing. After I soloed at home, I did a mock checkride for my other instructor and he soloed me too. Having two instructors did cost me more in total flight time, but it minimized calendar time because instead of losing proficiency by having long gaps between lessons, I was gaining proficiency by flying regularly. It also helped that the two had different teaching styles, so if one wasn't getting through to me, the other would. Still, I wouldn't recommend it if there's any other option