What will happen if I moved to another country, but didn't change my store region?

AFAIK, if you can provide your citizenship documents, Valve let you keep your regional pricing.

good question

Absolutely nothing happens but you might have problems buying stuff if the address of the payment method doesn't match the country your Steam account is set to.

Real case here, but I am not permanently migrated per se.

I am a Malaysian that are staying and working in Singapore. I go back to Malaysia 3 to 4 times a month.
Whenever I access Steam store page, they are showing me Singapore's store.
But when I login, the store will turn into Malaysia store.

I still buy games using Malaysia region price (and yes, when I'm logged in, "Fight of Gods" is unavailable for me).
I do own a Malaysia credit card, so I don't really need to change my Steam region.
All my games are still accessible.

I do own a few region-locked games that my trader friends gifted to me.
I can play the game just fine, but I am not able to access the store page.

In my case, I'm spanish and I was studying in Italy for 4 years only returning to my country on Christmas and Summer and I hadn't any problem when buying games on Steam Store page (and my credit card have a spanish adress) or activating keys. I suppose that like this two countries have the same prices (and are full members of the EU) and haven't a ban policy similar to Germany there are no problems, but if you move to a too much different country than yours there could be problems.
I hope it helps you and good luck with your life in a new country ;D