One slipped slate may be a local fixing failure
If the courses around it are secure, a roofer may only need to re-fix or replace the one that has moved.
Repeated slips are different
If slates keep coming loose from different parts of the same slope, the fixings across that area may be deteriorating. Simply replacing each slate as it falls can become a losing battle.
Thickness and overlap matter
A replacement slate needs to lie flat with the surrounding roof and maintain the correct headlap. A close colour match is useful, but fit comes first.