When lifted frame homomorphisms are closed

It is known that any frame homomorphism into a compact regular frame is closed. So the lift hβ:βL→βM of any frame homomorphism h:L→M to the Stone–Čech compactifications is closed regardless of whether h is closed or not. Here we study conditions which ensure that the lifts hλ:λL→λM and hυ:υL→υM to the Lindelöf and the realcompact coreflections, respectively, are closed.