First of all, you cannot bring the IP address along as you move to another hosting provider. IP addresses can only be moved between hosting providers, if you have a block of addresses, which was large enough to be announced over BGP.
Any approach you take to moving would require a transition period where you are using one IP at the current hosting provider and one IP at the new hosting provider.
If you are hosting only the website and not DNS for the clients, then the clients will have to update their DNS records, wherever they are hosted. For subdomains, this could be avoided, if the customer was using a
CNAME record. The
CNAME record has a tiny additional overhead to look up, but that is rarely significant. A
CNAME record could be used for
www.example.com, but it would not work for
example.com itself, since it would have to be the only record, and
example.com would have a
NS record and possibly
MX record as well.
So in that situation there is really no way around coordinating the transition with the customers.
If you were hosting DNS for your customers as well, they could have their
NS records point to names inside your own domain. That way they don't need any glue, and their domain will have no IP addresses other than on the DNS servers you host for them. As with the
CNAME records, this has a tiny additional overhead.
This approach however does require you to host their entire DNS zone, and you would have to be hosting it at two or more different hosting providers to provide high availability.