I can't sort out what these different participle phrases are called, or if they are adjectivals or adverbials
an adverbial has freedom of mobility within a sentence, but some adjectivals may be pre, or post modifiers
examples
The [running] man has arrived
The man[, running,] has arrived
[Running down the road,] the man was singing
The man[, running down the road,] was singing
The man was singing[, running down the road]
I saw Bill [running down the road]
All of them modify the man, not running, so they are adjectival.
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All of them modify the man, not running, so they are adjectival.
There's more to it than simply saying they are adjectives. In fact none of them are adjectives and they are not all modifiers:
[1] The running man has arrived.
Running” is a verb phrase modifying “man”. It's the same as “a sleeping child”, “the condemned man”, "the gleaming showroom".
[2] The man, running, has arrived.
A gerund-