come (do)ing vs. come to (do)

Examples:
come to help = come helping ?
come to visit = come visiting ?
come to look for = come looking for ?
come to (do) = come (do)ing ?
, etc.

You can only use the ‘-ing’ form with certain verbs that have a motion component, so ‘come helping’ is not possible, and we cannot generalize to ‘come doing’.

Excluding “come to help” and “come helping,” is any of the examples interchangeable? If none, what is the difference?

The form with the infinitive is a verb phrase of purpose; the form with the -ing verb is a verb phrase of ongoing activity.

Huh…, that sounds a bit too linguistic to a language learner like me.

Could you be so kind as to put it in plain English?

The difference between them is minor and they are often interchangeable. However, the difference is:

come to help/visit, etc. = indicates purpose (I have come for the purpose of helping/visiting, etc.)
come helping/visiting, etc. = indicates an ongoing activity (I am helping/visiting now. etc.)

come to visit = come to a place with the goal of visiting

come visiting = coming + visiting. ‘Visiting’ describes the activity of the coming.

I see! Thanks a lot!

oxfordadvancedlearnersdictio … onary/come
come doing something (+ adv.preposition)
to move in a particular way or while doing something else

  • The children came running into the room.

learnersdictionary.com/search/come

  • The dog came running when she called it. [=the dog ran to her when she called it]