A good question with something of a legalize answer.
All your exiting purchases made in the first country remain yours to use , however game centre type games and software updates require you to be logged in with your Apple ID and associated to your current billing address and payment details.
To comply with the EULA license when you change country you need to update the regional app store and change your details to reflect the new country local billing address and a local payment method.
Now here is the but ! - if you retain your first country bank and credit/debit card these have regional codes (First four normally) and Apple will regret it in the new country !
So you need an alternative payment method (Local bank card) or gift card and in some locations Click to buy (Charges made are linked to other utility accounts such as mobile or landline telephone providers).
Now here is the thing many think Apple use IP address location services to prevent cross border purchases- They don't but rather use the billing address and payment details.
So if your move is temporary (Education, Temporary Work posting or Military posting ) and your first country address and bank details remain- then simply continue to use these - you can download from just about anywhere with good Internet speeds.Not approved but works seamlessly !
If your move is permanent and you have a new billing address and local payment method then certainly change the store and update the account details to the new country.
As for warranties - With the exception of the iPhone all other Apple products have global warranties.