I’ve changed my mind. The –ing form used in the sentences is in fact the Present Participle. The problem I had at first was the notion that the subordinate clauses were functioning as adjectives. I couldn’t see that, therefore, I thought they had to be gerunds in appositive or relative noun clauses or something like that.
I then found a great explanation that I think sheds more light on the use of the Present Participle in the sentences in question and as I initially suspected, it is not acting as an adjective:
To replace a sentence or part of a sentence:
When two actions occur at the same time, and are done by the same person or thing, we can use a present participle to describe one of them:
• They went out into the snow. They laughed as they went. They went laughing out into the snow.
• He whistled to himself. He walked down the road. Whistling to himself, he walked down the road.
When one action follows very quickly after another done by the same person or thing, we can express the first action with a present participle:
• He put on his coat and left the house. Putting on his coat, he left the house.
• She dropped the gun and put her hands in the air. Dropping the gun, she put her hands in the air.
The present participle can be used instead of a phrase starting as, since, because, and it explains the cause or reason for an action:
• Feeling hungry, he went into the kitchen and opened the fridge.
(= because he felt hungry…)
• Being poor, he didn’t spend much on clothes.
• Knowing that his mother was coming, he cleaned the flat.