Allow admin to define REPORTING PERIOD & frequency.
Example:
MONTHLY, as of the last day of the month
Weekly, as of every Friday

Base reminders to employees on the reporting deadline.
There's no point reminding people that they have expenses that haven't been submitted (say on the 12th of the month), if they are NOT ALLOWED to submitt them until month end.

Many companies I've worked for only process expenses Monthly, or for the same periods as the pay period.