How frigates, cruisers, and destroyers changed roles

Did sailing frigates evolve into battleships, leaving later cruisers to inherit the older frigate role?

Broadly yes, although disruptive technologies create new types that later tend to settle into capital-ship or cruising roles.

The broad lineage separates the capital role of ships of the line and battleships from the dispersed work of frigates and cruisers. As the sailing frigate evolved toward the battleship, it left a cruising-role gap later filled by cruisers. New systems can disrupt that pattern, but their ships often settle into one or both older roles. Destroyers began as torpedo-boat destroyers and increasingly assumed frigate-like duties as cruisers moved upward in capability. Navies can therefore end up with several tiers of cruising vessel rather than a stable boundary among frigate, destroyer, and cruiser.