Former British colonies and the US itself will be the worst.
They all share the British approach to quarantine of animals
(30 days) and take is very very seriously. In any event, have all your papers and vet records along. maintain all shots and treatments as you would at home. Also check on bringing your dog back home too. Customs
laws work in both directions. Your dog is not a citizen! Getting back home can be a real problem. Find out what you need to have in detail for every place you go.
Going ashore for vet visits can be a serious problem too. There is no such thing as an excuse to by pass customs
rules - if caught.
The odds of being caught are not huge but if you are caught I don't think any country will let you off easy at all and you could lose your dog on the spot. It's not something to fool with. With recent world events
concerning livestock many countries are more interested in enforcing customs laws than they used to be if they were not already.