Parts of Speech of: come+ verbing

Question

He came walking

He came waving both hands

He came yelling

And

They left crying

Do you parse walking, waving, yelling and crying as verbs? If not, what would you parse them as?

Edit: crying added

Answer 1

They are verbs, of course--present participles. Waving, for instance, takes a direct object, his hands. Syntactically they are (take a deep breath before you try to say this)

• subject-oriented predicate adjectivals

That is: they play the role of adjectives in describing a noun; the noun they describe is the subject; and they occur in the predicate, among the dependents of the verb, rather than being included as attributive adjectives in the subject.

Exactly the same thing can be done with ordinary adjectives and adjectival past/passive participles.

He came home wounded.
He came home elated by the news.
He came home happy.
He came home naked.

None of these describes the quality of the movement: they describe the subject.

There are also object-oriented predicate adjectivals, which likewise may be ordinary adjectives or present or past participles (with or without dependents):

I drink my coffee black.
I drink my coffee steaming.
I drink my coffee loaded with whipped cream.

Answer 2

I'm not sure I'd parse those as adverbs as the comments imply. It's certainly a possibility, but as a native American English speaker when I think present participle derived adverbs, I tend to think of words like "interestingly".

I might instead parse it as a past progressive verb form, understanding it to mean, for example, "they were crying when they left" or "he was yelling when he came".

Not sure, though.