Who is responsible for airspace separation when an IFR flight is cleared for a visual approach?
Visual approach responsibilities
§5-5-11
Pilot's end:

(3). The pilot must, at all times, have either the airport or the preceding aircraft in sight.

Controller's end:

(3). Provide separation except when visual separation is being applied by the pilot.
(7). Continue flight following and traffic information until the aircraft has landed or has been instructed to change to advisory frequency.

—AIM

For coordination with/between nonapproach control towers
§2-1-14 Coordinate Use of Airspace

(a). Ensure that the necessary coordination has been accomplished before you allow an aircraft under your control to enter another controller’s area of jurisdiction.
(b). Before you issue a control instruction directly to a pilot that will change the aircraft’s heading, route, speed, or altitude, you must ensure that coordination has been completed with all controllers whose area of jurisdiction is affected by those instructions unless otherwise specified by a letter of agreement or facility directive. If your control instruction will be relayed to the pilot through a source other than another radar controller (FSS, ARINC, another pilot, etc.), you are still responsible to ensure that all required coordination is completed.

§2-1-16 Surface Areas

(a). Coordinate with the appropriate nonapproach control tower on an individual aircraft basis before issuing a clearance which would require flight within a surface area for which the tower has responsibility unless otherwise specified in a letter of agreement.
(b). Coordinate with the appropriate control tower for transit authorization when you are providing radar traffic advisory service to an aircraft that will enter another facility’s airspace.
(c). Transfer communications to the appropriate facility, if required, prior to operation within a surface area for which the tower has responsibility.

§2-1-17 Radio Communications

(a). Transfer radio communications before an aircraft enters the receiving controller’s area of jurisdiction unless otherwise coordinated or specified by a letter of agreement or a facility directive.

—ATC (JO 7110.65W)

Short version of the above
ATC (terminal or tower) will need to instruct an altitude and/or heading so you don't deviate and surprise/buzz a Class D you're not going to. Otherwise they'll need to transfer the comms, which I'm sure they don't want to. Unless there's a prior agreement between the different facilities.
Pilot needs to have preceding traffic or field in sight.
Terminal ATC needs to keep an eye on all traffic and provide traffic information, until they handoff the pilot to tower or CTAF.