I found the following etymology of the word "ambassador" on Wiktionary.
From Middle English ambassadore, from Anglo-Norman ambassadeur, ambassateur, from Old Italian ambassatore, ambassadore, from Old Occitan ambaisador (“ambassador”), derivative of ambaissa (“service, mission, errand”), from Medieval Latin ambasiator, from Gothic (andbahti, “service, function”), from Proto-Germanic *ambahtiją (“service, office”), derivative of Proto-Germanic *ambahtaz (“servant”), from Gaulish ambaxtos ("servant"; also the source of Latin ambactus (“vassal, servant, dependent”)), from Proto-Celtic *ambaxtos (“servant”), from Proto-Indo-European *h₂m̥bʰi-h₂eǵ- (“drive around”), from *h₂m̥bʰi- (“around”) + *h₂eǵ- (“to drive”).
I'm wondering how Gothic (andbahti) became Medieval Latin ambasiator phonetically, especially how h in andbahti became s in ambasiator. I have read some books but still couldn't find a specific reason for this sound change.
Here is my personal explanation of this sound change:
This is a sound change of partial assimilation. h changed into s retaining its own fricative feature and gaining the alveolar feature from t. Then epenthetic vowels (between s and t) were added.
Still, I am not sure if my the explanation is correct. If yes, are there any other sound changes similar to this one?
Yet, I am also not sure whether h actually changed into s historically.
Edit: Thanks for the answers below, really appreciate them!