Each Ride Series has a set end date, which will often be set to the end of the school year for the rider. After the end date, no new rides in the series will be available for you to claim as the Primary CareDriver. Any new rides for the rider will be scheduled as a new Ride Series that can be claimed by any CareDriver.
If you're interested in claiming a Ride Series with a rider you've driven previously, you can find more information here: Can I Claim the Same Ride Series I Had Last Year?