Which visas are checked by the airline when checking for the first leg of a RTW?

Naturally, I expect immigration at each arrival airport to check that you have a valid visa for entry into their country, unless such requirement does not apply between the pairs of nationalities involved. Related documents sometimes need to be presented at that point, which may include onward travel.

Our the last time we booked a multi-country but not RTW route, the person at the checking counter invoked assistance from 3 more employees and spend over 90 minutes with us until he finally was convinced that we were at minimal risk of have a visa-on-arrival refused. But this was for the second stop only, so I'm worried if there were 10-12 stops further out.

How does one convince the check-in agent that the rest of visa will sort themselves out. There are some continuous which issue Visa-On-Arrival (obviouusly based on certain criteria, so it's not 100% sure) or some visas may be obtaining quicker while visiting neighboring countries as part of the RTW trip. For example, the Brazil visa obtained in Iguaçu takes 24 hours compare to weeks for the usual one.