Using a different tense in a dependent clause?

No, you shouldn't change tenses in the middle of a dependent clause.

To quote @dm_with_secrets, there are no rules - that is, nothing is strictly illegal or not allowed when you're writing. But that being said, changing tenses in the middle of a dependent clause is poor grammar.

Take this sentence:

Matt typed out an answer on StackExchange, as his partner watched television on the couch.

These two verbs are both past tense, so the clause makes sense. However, this sentence becomes grammatically incorrect if you changed either of these verbs to present tense.

Matt typed out an answer on StackExchange, as his partner watch television on the couch.

This sentence doesn't make sense anymore - I can't be doing something in the past tense while the other person is doing something in the present tense.

To better demonstrate this, let's replace the second verb with a participle. "-ing" forms of words (running, walking, etc.) are called participles. You can, grammatically, change from a regular verb to a participle verb in a dependent clause, but the tenses still have to match.

This still makes sense, because you have a past tense verb and a past participle:

Matt received a parking ticket this morning, because he was parking illegally.

But this doesn't make any sense:

Matt received a parking ticket this morning, because he parking illegally.

Normally, when the main clause is past, verbs in any dependent clause are backshifted (to past, unless they were already past, in which case they will usually be shifted to past perfect).

There is an exception, where what is stated in the subordinate clause is timeless, or still continuing:

He said that he was/is living in London now.

If he is still living in London now, then either is or was is possible.

But your example sentence would be very unusual: the only reading that makes any sense is that people have continued to slam the doors right up to the present.