If I first publish my website from my server which is in city A, and then move this server to city B (which I expect to change its IP address), will the website's users still be able to access it under the same URL? I'm thinking about this problem from the point of view of DNS - I expect to be able to change the mapping of this (theoretical) website's domain to the new IP address with the domain registrar, but I'm worried that some users which have visited the website prior to the relocation will be pointed to the old IP address.
I've heard that DNS mappings are "cached" and that this cache expires only after some time which could lead to the pre-relocation users not receiving the new IP address information, but I was unable to find any information specific to the aforementioned case.
To summarize, do I have to worry about the pre-relocation users not being able to access the website through the same URL after the relocation and the change of the IP address? If yes, then what could I do about it?
(Side question: Would an ISP allow me to retain the same IP address after the relocation to the new city? What about moving to a different country? I'm asking this because if I could retain the IP then any trouble with the cache or whatever would avoided.)