Is it possible to clear DNS cache globally?
Solution 1:
The short answer is no.
The reason behind this, is because just flushing DNS (a restart should do this) won't help if the local PC and browser cache the DNS as well.
The option is to restart your DNS and ask users to reboot. It's a little overkill, but if they don't understand DNS cache, you will be spending enormous energy on helping them clear local cache...
The best solution to ensuring that this doesn't happen in future, is lowering your TTL for the domain a few days before moving to something like 5 minutes (1 if you're confident you have low levels of DNS queries for that domain), changing it, and then upping the TTL afterwards.
Aside from that, you could also create a redirect on the website (iptables, modrewite) to point to a new IP.
Solution 2:
It's difficult to get in contact with administrators of the ~32 million DNS servers out on the public Internet much less get them to clear the cache on their servers without a really compelling reason, like buying them all a drink for their trouble.
Proper planning ahead of a change to drop the TTL down is best-practice. Too late for you, but at least now you know.