You can send an email with the Emailer that includes the "Swap requested positions" token. Limit the date range in the token options to the time frame of your choosing (e.g. the coming weekend), and specify to not send the email to anybody who can't fill any of the swaps. If you send out such an email on Thursday that will serve as a reminder for all the pending swaps on the coming weekend. Also, you can optionally turn on the option to include links that allow volunteers to accept the swap via the Web Terminal so that they can click on a link in the email to accept the swap request.
Note: With the Online and Plus editions of MSP, you will find a Auto Reminders pane in the Web Terminal Settings window, where you can set up an automatic email to go out to volunteers and include any open positions for the coming week.