There are a few different methods to control when and if the ability to book expires.
Let's go through them step by step. Note: for all items, we will be working in section Admin Panel » Setup » Plans.
Go to the section Plans » Time Passes. When you are setting up day passes associated with a plan, you have the option to provide a number of free passes that expire either weekly or monthly. This will be 1 week or 1 month from the purchase date of the plan.
Go to the section Plans » Bookings.
When you are setting up the bookings associated with a plan, you have the option to provide a number of free hours associated with particular resources. These expire/refill to that amount at the conclusion of each monthly billing cycle.
Note that you can also provide different amounts of credits for different resources.
Go to the section Plans » Bookings. An alternative to including credits with the plan purchase price, you can offer your members the option to purchase passes at a discount in this section. Note: these passes do not expire.