Not automatically, for RapidSSL. For other vendors and/or certificate grades, it may be be automatic.
RapidSS don't automatically invalidate a certificate when it is re-issued, according to their Certificate Practice Statement. This would be a function of how much you pay for the certificate.
In section II.B.5 of the currently-google-able CPS:
GeoTrust will not revoke a Certificate previously issued following a
refund or reissue request. A request for a refund or reissue of a
Certificate will not be treated as a request by the Subscriber for
revocation of a Certificate previously issued by GeoTrust unless the
Subscriber follows the procedures for requesting revocation as stated
at Section III.I. of this CPS.
In section III.I it says:
Certificate revocation is the process by which GeoTrust prematurely
ends the Operational Period of a Certificate by posting the serial
number of the Certificate to a Certificate Revocation List. A
Subscriber shall inform GeoTrust and promptly request revocation of a
- whenever any of the information on the Certificate changes or becomes obsolete; or
- whenever the Private Key, or the media holding the Private Key, associated with the Certificate is Compromised; or
- upon a change in the ownership of a Subscriber's web server. Subscriber shall state the reason(s) for requesting revocation upon submitting the request.
Elsewhere it minimally promises that revoked certificates will be added to the CRL 'at least once a week'.
Reading the Certificate Practice Statement of any purchaser of SSL certificate services is a good thing for a buyer to do.