Halliday & Matthiessen (2014: 500):
The non-finite variant corresponds to the dependent enhancing clauses with conjunctive preposition; e.g.
death [[ × by drowning]]
a pain [[ × like having a red-hot needle stuck into you]]
Blu-ray: death [[ × by streaming]]
The trouble [[ × with predicting climate change]]
Children need help [[ × in learning to control their emotions]] .
In Seoul, there seems to be anger [[ × at being taken for granted as an American satellite]]
Since the noun functioning as Head is generally the name of a process or property, these often have close hypotactic parallels, e.g. he was angry || × β at being accused; if you help me || × β in cooking the dinner; it’s difficult || × β with everyone having a part.
The non-finites could in fact be reworded in the same way as the finites; e.g. the trouble with everyone having a part as the trouble [[[ = which arises || × β with everyone having a part]]]. But there is no need to treat either kind as other than embedded enhancing clauses.