I finally gave up trying to answer this myself and took the problem over to ELU. As I hoped, John Lawler came through.
The construction has apparently not been named, but JL suggests treating it as
a serial verb construction (SVC) of two or more verbs in sequence, like go swimming or Come help me! He describes the construction in your example as [my emphasis]:
... the initial verb part of the serial verb is
- an intransitive motion verb (go, come)
- a transitive causative/inchoative of a motion verb (respectively: take, bring)
- a transitive verb that entails some kind of induced motion (pluck, send, toss, drop, etc.)
while the gerund* part normally describes some property of
- the motion induced by the verb (lurching, zigzagging, spinning) [OR]
- the object or person caused to move (muttering, shuddering, screaming, unmoving)
In either case, it is the moving object that functions as subject of the gerund constituent.
* I believe he uses gerund in the sense CGEL uses gerund-participle.
He goes on to remark that this construction is 'discontinuous', like a "phrasal verb" where the second part of the verb comes after the object. Thus sent it zigzagging is analogous to do it over or ask her out.
Fascinating stuff. There seems to be a substantial literature on SVCs, but mostly about languages other than English (including your own). I did find this article; section 2.1 is most relevant, but the conclusion is of great general relevance.
Do look at JL's answer for more detail. And I think you will particularly enjoy his following Comment about Harry Potter.