In some language with a moderate case system, whatever case can be used for complements goes on the infinite to form an adverbial/complemental participle. Over time, the participle is reanalyzed and can be used as a full adjectival participle, both in attributal, complemental and adverbial ways.
An overview of this process, with a Slavic influence, would posit that instrumentals are used on some complements. This is especially useful, as it lends itself to two developments:
1) "by [infinitive]", i.e. marking the infinitive to show the manner in which something was carried out.
2) "as (implicitly: doing) [infinitive]", i.e. marking the infinitive to show that the infinitive is a complement telling us something about the subject (or possibly object).
The second option easily could include elliptical uses of subclauses (X who is verb-instr. -> X verb-instr) and soon, the participles may be very much like classical participles for those.
I am thinking of doing something similar for that case system with regard to how participles came about. In that, the case markers that form present participles will be closely related to the complements and the instrumental-comitative.