They stand in contrast to adverbial adjuncts, which can be removed from a sentence without altering its structure or meaning.
In this theory, adverbial complements are stored in the lexicon as part of the grammatical competence relating to the verb.
Here, adverbial complements are stored in the grammar as part of the caused motion construction which is a sign in its own right.
Another construction-based theory combines the two arguing that certain senses of verbs co-occur so frequently with certain argument structure constructions, that the argument structures are also stored as part of the grammatical competence relating to the verb.
So, in the case of put, in accordance with this theory, adverbial complements are both part of the argument structure construction and stored as information regarding the verb itself.