This paper studies expiration-based caching systems in which caches assign a timer to each content they store and redraw the timer upon a cache miss. The modern Domain Name System (DNS) hierarchy is a valid application case and will be used throughout the paper. We introduce analytical models to study expiration-based caching systems based on renewal arguments. For polytree cache networks, we derive the cache performance metrics and characterize at each cache the aggregate request process, the thinning process and the miss process. A constant TTL policy is proved to maximize/minimize the hit probability if the requests' renewal function is concave/convex. We find that no distribution maximizes the hit probability anywhere in a network of caches. We validate our theoretical findings using real DNS traces (single cache and network cases) and via trace-driven simulations (network case).