With no professional linguist afoot, this amateur asks y'all's indulgence:

We gotta do some parsing here. Or diagramming. Verbs that sometimes dress as nouns are called verbals. In the structure "you don't like my running," for example, the present participle running suddenly finds itself changed into a gerund (part of the verbal family), the direct object of the sentence, and as such, it needs an adjectival modifier - in this case, the possessive pronoun my.
Now let's take the verbal and make it transitive, as in the example Wow provided: "You don't like my running this equipment." Equipment becomes the direct object of running, which, while it hasn't forgotten its verb roots and functions as one within the expanded set ("running this equipment"), it still becomes part of the direct object of the entire sentence, needing "my" to modify it in the same way.

I am not certain of all the terminology; it's been years and years. But I figure if I don't know it, a lot of y'all might not either - this was my best shot at explaining this structure.

Her leaving misplaced and dangling modifiers for another time will probably be appreciated by kindred spirits who find over-long posts as tedious as she does