Before the war (1939) there was a big movement to get people to travel and see the countryside: https://en.wikipedia.org/wiki/Strength_Through_Joy

After the war started, it was war conditions: freedom of movement was curbed. Checkpoints were everywhere and papers were frequently checked when traveling.

Jews, and then others were quickly banned from even using travel agencies (let alone cruise ships, normal trains, etc), so other than for Germans, travelling quickly became difficult to impossible.

I can't find a ton of stuff on occupied populations in the 3rd Reich. But basically, no, Czech could certainly not cross over into Slovakia, who was suddenly independent, unless in very good standing with the Nazi Government of Bohemia and Moravia. Everyone was basically supposed to stay put, they definitely didn't want refugees moving about freely. Everyone was (and still is) registered to their apartment, and it's up to the government to get that changed.. so just up-and-moving was technically illegal and could be stopped.

The other factor is that it was war time and people were on rations. The infrastructure was used for war, not leisure. Few could afford it, but yes, the Nazi elite had their pick of French chateaus, etc to visit. They also occupied all the historic sites of occupied areas. Taking up residence and setting up offices in castles and monasteries, etc... so not open to the public anyway anymore.

But before the way tourism was flaunted on the world stage. Hitler made museums free and had huge beach-resort projects started on the North Sea coast (all of these were shelved in '39)

Toward the end of the war it's been noted how free movement was... Czech doctors from Prague went to Dresden to care for the wounded, and there was simply no-one at the border to stop them. Think of the millions of refugees heading West away from the Russians; until the Russians (and the rest of the Allies) set up shop and started restricting movement again.

tl:dr borders were enforced. You couldn't just go from an occupation zone to the Reich proper without being sent there.

Sorry about the lack of sources, I'm at work. But I can find sources for most of this (some are anecdotal speaking to folks in Prague that were old enough to remember)