Installation and retrieval of pressure control device releasable assembly

A well system can include a pressure control device including a releasable assembly, and a running tool including a lock mechanism operable in response to a change in a fluid flow rate through the running tool. A method of retrieving a releasable assembly of a pressure control device can include engaging a running tool with a latch, applying a force from the running tool to deactivate the latch, and changing a fluid flow rate through the running tool. A pressure control device can include a releasable assembly having a passage, and a latch releasably securing the releasable assembly in an outer housing, the latch including a piston having set and unset positions, and the piston being displaceable toward the unset position in response to pressure applied to a port in the outer housing, or in response to displacement of a profile disposed in the passage.