I would call it a “Journey” Dog

Man you are just ripping off my welcome mat.

I’d say a dog would still go to predictable places eg. they’d go for a walk on their usual route (maybe a bit larger).

Not all those who wander are lost. Sometimes they just play dog roulette

Might get kind of tricky when crossing a busy street

As long as you only want to go to places that smell of piss

You may not end up where you meant to go, but you’ll end up somewhere interesting