A word or phrase for "going past a deadline"

The word you are looking for is miss. "miss the deadline"

This is how I would put it:

Participants are often being implored to finish their tasks by the end of November, as planned, and under no circumstances to let them drag on into December.

To underscore the urgency of timely completion of the task, I might also send reminder messages to each participant, specifying a "drop-dead due date" for each task:

The drop-dead due date for completion of your task is November 28. Please note that the task must be finished by that date; the intricacies of the project do not permit any extensions beyond November 28.

Of course, if you say that, you really shouldn't allow any extensions beyond the drop-dead date.