The article says he would “decamp” to Mexico at times and then pick up where he had left off. I read this as he’d take a break and fly (presumably) to Mexico and then fly back to where he left off.

Depends on what you mean by “unbroken”.

I’d assume that when he ran into legal issues like after crossing the Bering Strait the authorities would transport him via a car or bus so it’s not like he never took a motorized transport.

People always assume he's been walking continuously for all these years, and it's understandable because the headlines are always written to imply that.

He doesn't. He walks some, marks where he stopped, takes a break, continues from where he left off.