Pronouns + Gerund Form

Someone linked 'Seven Bar Jokes Involving Grammar and Punctuation' by Eric K. Auld in a language exchange forum, of which I personally like three and four, and after some discussion about the second one in particular, I made some interesting discovers about English grammar.

'A dangling modifier walks into a bar. After finishing a drink, the bartender asks it to leave.'

 Without a dangling modifier, the latter sentence would read as:

'After finishing a drink, it was asked by the bartender to leave.'

One person then ask would it be possible to add it as in, 'After it finishing a drink', citing the people's use of 'After him finishing'. Now the knee-jerk reaction was no, 'After it finishes a drink' is correct. So I decided to research further into it.

I stumbled upon a post by Jakub Marian on the 'pronoun + gerund' pattern, the very pattern we are seeing in the joke in question (note: as a prepositional phrase not a gerund). The summary is that traditionally, the accusative or direct object pronoun before the gerund is incorrect and the genitive or possessive form before the gerund is correct, however now it's become part of the vernacular so both are correct. Additionally, it may come across as confusing to use the traditionally correct possessive as in, 'They objected to the youngest girl’s being given the command position', where it could be misheard as 'several girls being the command position'.

Another resource, an answer on Stack Exchange by nohat and RegDwigHt, goes into specifics, referencing the Cambridge Grammar of the English Language. The 'gerund clauses' can accept the accusative (direct object him) or the genitive (possessive his) as subject, however 'present particple clauses' can accept the accusative (direct object him) or the nominative (subject him) as subject, accusative being less formal in both situations.

Thus, 'After it finishing a drink, the dangling particple was asked by the bartender to leave.' is a perfectly grammatical sentence.

I would wager that the reason 'After it finishing a drink' sounds as if it should be conjugated is because with it the nominative and accusative case are the same: 'It eats it' versus 'She eats her'. There's no distinction, thus a native speaker defaults to this phrase to sounding as a clause with a subject (reading it as a nominative) rather than as a present participal clause within the prepositional phrase (reading it as either nominative or accusative, accusative being more natural in my opinion).

Some further reading on phrase types.