sandyRoe - //If the support and funding stopped at age 18, say, wouldn't there be an incentive to show the youngster the door?//
If you regard your foster child purely as a cash cow, then yes.
Hopefully the vetting system weeds out such people, and foster families have enough emotion and love and care invested in their relationship that would mean that their family member is not required to vacate the premises on attaining the age of majority.