Consistent verb tense

Hello,

I am curious as to whether the following sentence has the correct verb tense (i.e. simple):

This is what we say to a guest who leaves a party we invite them to (i.e. Thanks for coming).

The verbs say, leaves and invite are all in the simple present because it's used to discuss something that happens as a routine. When somone routinely leaves a party, we routinely thank them for coming.

However, I am most curious about the verb invite. Specifically, should it be invited because it was a past action or is it invite because verb tense shifts in a sentence are grammatically incorrect?

Many kind thanks to anyone who can answer this question.