I was always taught that if you can replace a participle with a pronoun, then you should use the possessive case. So instead of "I saw him eating", because I can substitute "it" and say "I saw it", I should say "I saw his eating." "His eating" is the object here, and it can be moved around as such. For example, I might say "His eating is good," but it would be incorrect to say "Him eating is good." This rule of "Can I substitute 'it'?" has proven helpful in most cases.
That said, when I come to very long participial phrases, using the possessive can seem awkward. For example, even "I saw him eating a donut" sounds much more natural than "I saw his eating a donut", even though I can still substitute it for the entire phrase and say "I saw it." But if "I saw him eating a donut out on the front lawn yesterday morning with a friend," then it seems even more awkward to use the possessive, even though it still seems more grammatically correct.
Any thoughts on this? What is the rule? When is it proper to use a possessive in combination with a participial phrase, such that the entire thing acts as an object, and when is it proper to use the phrase to modify the object?