Is it unusual for a professor to submit recommendations to other universities more promptly than their own? I am applying to graduate programs at several universities, including the one that I am currently at.  My professors sent their letters of recommendations to the other universities promptly and quickly, but have not sent them to their own university yet, despite sending them at least over 5 automated requests.

I am highly worried because the deadline is coming up in two days.  Is this a common occurrence, and what I should do about it?
There's not enough information to say for sure, but one possibility is that they know the deadline isn't strict because they've seen what happens from the inside.  When submitting a letter to another university, it's hard to know how harmful a delay might be, but your recommenders may know that in your department nobody even plans to look at the letters until the end of next week.  (It's not uncommon to schedule a buffer period between the deadline and the beginning of organized review, precisely to give late letters a chance to arrive.)  Or they may trust that their colleagues on the admissions committee will forgive them for being late.