I saw a shared video of someone with a buddy with catch assist. It seems like a normally wasted ball (landed during the 'attack' of the wild Pokémon) is being headbutted back, delivering a 'Great throw' (I personally think it would be an awesome combo). Are the normal catch chances applied? Is this assist always there? Does the original circle matter for the type of throw, or would it always be great?
Are the normal catch chances applied?
Yes, the normal catch chances still apply. Pokémon can still break free if your buddy throws the pokéball back, so it's safe to assume normal catch rates still apply (source needed).
Is this assist always there?
No, this does not always apply. Much like Critical Captures in Pokémon Go, it is a random chance for Catch Assists to trigger.
Does the original circle matter for the type of throw, or would it always be great?
It will not always be a great throw. There is evidence for Nice and Excellent throws as well. Based on my experience and other threads I’ve seen as well, the catch bonus (if any) will typically be based on what the original throw would have been.
So if you throw a Pokeball and it would be outside the circle, you likely wouldn’t receive a catch bonus despite it being knocked back into the center. Otherwise, it would be generally be nice, great, or excellent relative to the size of catch circle.
Curve ball bonus' also still stick! If you apply a curve throw, miss, and your buddy throws the pokéball back, you will be awarded the circle ball XP bonus if the Pokémon is caught