In her later years, my mil fostered a series of older dogs whose owners were ill or who needed care themselves. Some only stayed with her for a month or two until their owners recovered, some stayed longer until they themselves died.
You do have to be able to live them and then lose them, the same as human foster parents but I know it gave her much satisfaction.
She had the assurance that if she became unable to look after the current dog, the charity would take the dog back.
It was a local independently run charity but the Cinnamon Trust sounds very similar.