help me find an English grammar rule?

In English with some verbs you can use their ing-form after go, i.e. sentences like "I go running often", "we are going shopping" etc., but with other verbs this is not allowed, i.e. you don't say "we are going eating"(*) but "we are going (out) to eat".

I think the rule is that the construction is only allowed with movement verbs, like go walking, swimming, dancing, etc. all work, but not with reading, knitting or painting. I'm actually unsure about playing, but I think not? OTOH working and hunting seem okay in the construction?

I tried finding the rule for this in grammar explanations but I'm not even sure whether the -ing is considered a gerund or a present participle here. So I was hoping that maybe the English language geeks on my f-list could point me.