How does Apple Watch Activity app deal with timezone changes?
I love traveling but I have the same problems with my appleWatch.

I contacted the Support, It's a well known problem since the watch have launched. The "Bug" isn't fixed yet.

The Genius told me, the only way to handle that problem is to deactivate the automatic time adjustment of the iPhone and Watch. 

Well it's not a satisfying solution but it works. The only problem is that you need to calculate the current time as it won’t display the correct local time.