"Catena" is a good start, but a more specific term for this--since "catena," as far as I am aware, can refer to all sorts of strings, not just of verbs--is serial verb construction. ("Verb serialization" and "verb stacking" are other names for this phenomenon.)
The Wikipedia article on serial verb constructions specifically cites Mandarin as a language in which they are to be found. It adds this note:
In Chinese, however, there is often no clear distinction between
serial verb phrases and prepositional phrases. The first three "verbs"
in the above sentence ("wǒ zuò fēijī cóng Shànghǎi dào Běijīng qù"/
"I sit aircraft depart Shanghai arrive Beijing travel"/
"I travel from Shanghai to Beijing by aircraft") may alternatively be regarded as prepositions
(this applies particularly to words like cóng which do not normally
appear as independent verbs). Words used in that way in Chinese and in
some other languages are commonly referred to as coverbs.
As for the question of whether they exist in English, Wikipedia has this to say:
Certain expressions resembling serial verb construction are found in
English (surviving from Early Modern English), such as let's go eat
and come live with me.2 In such constructions, the second verb would
normally be regarded as a bare infinitive (and can generally be
replaced by a "full" infinitive by the insertion of to before it).
Cf. compound verb/complex predicate, a related-but-different idea