@warwick I’d have to agree with @insaneHoshi here.
The Example of Romania on page 8 of the rulebook seems to address this too:
“If the roundel is not located in terrain, but you need to cross a terrain type in order to reach the roundel from the border of the land zone, you are subject to “border terrain” rules. Border terrain rules for that terrain type last one round”.
So while that specific example seems to be talking about combat, it does still mention that the terrain rules apply when crossing a mountain border. I just mention that one since it’s using an example just like what you asked, if you cross into a territory with a mountain border, but the roundel isn’t in the mountains.
But also with the rule just being “All land units (except cavalry) have their movement reduced to 1 when subject to Mountain rules”, I’d think you have to take that at face value to mean crossing a mountain border reduces movement to 1, even in non-combat movements.