I'd say it's not typical, as most liveaboards don't cover that much distance. But certainly possible. In the Caribbean there's a lot of islands that are not all that far apart and are different countries, and a few boats with regular or occasional itineraries that enter more than one country. I don't know if there are any similar situations in the Indo-Pacific region - I'd guess there are fewer anyway.
I think the only time we have done that was on a liveaboard leaving from Grenada in the Caribbean, and traveling through the Grenadines to St Vincent. It was a long time ago but I think the boat collected all our passports and took care of it for us. We might have even been asleep at the time. (That's a situation where a visa entering the country is usually a stamp given on entry to most visitors, rather than something that needs to be applied for in advance. If that was necessary the boat would hopefully let you know.) The Caribbean Explorer which travels St Maarten-Saba-St Kitts would be one. There have been some boats that depart from the US (Florida) to dive in the Bahamas. The Cuan Law, which usually stays in the British Virgin Islands, occasionally used to do a special charter that went down the island chain and hit lots of places. There's probably others.
