Can a family apply for a 6-month multiple-entry Thai visa while traveling, or do they need to return to the UK?

In your case I would not apply for any visa, it is time consuming because you got to stay the whole time of the application till approving in the country of application and you got to make a visa for every family member The application could take up for many weeks I guess you and your family members got a strong passport where you can go easy on most countries with the passport only For example Thailand allows you to stay for thirty days on Visa exempt and can be extended in the local immigration office for another 30 days, you can fly almost unlimited times back to Thailand ( max 180 day stay in a year/landborders non flight crossing limited 2 times a year) I would first check individual regulations of all the countries you want to visit and I guess in almost all you will not need visas I did similar trip long ago and than I only needed a visa for Vietnam where now it is not needed anymore The only thing you got to watch that you need onward tickets for the countries you want to enter on visa exempt . So in practice you already need to know when you go the next country when entering the one before and book those tickets

@Bent ************

Not