The "upcoming gigs" section of the dashboard will continue to display gigs after they have passed for a day or so for a couple of important reasons.
One is that many performers don't come home after a gig and immediately sit down to wrap up a gig in their CRM. Having the gig stay on the dashboard for a little while after the gig is completed keeps it front and center to remind you that you still might have some loose ends to tie up. It also prevents you from having to dig it up. "One click" access is very important to many of our users.
Another reason is that many gigs are not actually completed on the same calendar day as they are scheduled for. For example, if you are hired to do 3 hours of up close magic starting at 9:30 pm, the gig spans two calendar days. To allow for that, the gigs do not drop off at midnight on the calendar date they are scheduled for.
The gig, and the associated alert that suggests that you send a thank you and schedule a follow up with the contact, will both drop off within about 24 hours. After that, you'll be able to access the gig through the "Completed" section of the gig menu on the left side of the screen.