It is common to thank somebody for "a kind invitation". However, when somebody writes
I kindly invite you to event xyz,
does that not suggest that the person who invites the other is doing the invited person a favour? If yes, the meaning would be quite different from phrases such as
I cordially invite you ...
It is a pleasure to invite you ...
My google search produced many hits for the phrase "kindly invite", but all of them point to websites of non-native speakers/institutions.