Oh sorry, thanks for the correction on whose nameservers you're
using. But no, your domain name shouldn't lose them and your
domain shouldn't go offline, either.
Ideally when a domain registrar transfer is taking place, assuming
it's still using its current nameservers and nothing are being done
to the nameservers themselves, there shouldn't be any downtime
Note I said "ideally". In reality, anything can happen, especially
when technology is inherently flawed. (well, its creators are never
perfect in the first place, either. )
Other than that, you shouldn't have any problems. But again, if
you need to make changes to your domain, do them before or
after the transfer.
You should re-register them there - I doubt you will lose them or your sites would go offline because of the rapid changes in DNS that has been implemented. I would set them up immediately though to be on the safe side. And know where that functionality is before you transfer them
joephill, it's a good thing you asked that question because I just went to check my domain that I transfered between GoDaddy and BlueRazor that had custom DNS setup. The nameservers stayed on the domain but the registered IPs did not. Since I think the transfer only became effective this morning, don't think it was a huge deal - for me at least.
I reregistered the IPs but I don't think the site had a chance to go down because I just got notification from BlueRazor an hour prior that the transfer had completed. And I had checked the domain - for other reasons - at least once this morning and it was available. Maybe if you catch it fast enough, it won't be an issue. I don't know. I guess because it's registrar specific information it doesn't transfer with the domain - i.e. the stuff you don't see on a WHOIS search?? because my domain forwarding didn't transfer either but I didn't expect it to.
And now that I think about it, pay more attention to the transfer form and see if there's stuff that can be added. I honestly didn't pay that much attention to it because I didn't realize I was transfering the domain with custom nameservers in this particular batch. Maybe there's something in the form to keep this sort of issue from happening at the winning registrar you choose. I don't know, but it's a thought at least. Try contacting their support team to see if there's any assistance for that type of transfer.