I would not attribute the DCT as "fuel efficient" in itself. With standard drive cycles, DCT adds to fuel consumption about the same amount of fuel as an automatic transmission gear. (Both compared to manual switching gear.) Although the reason for the increase is different: DCT is continuously switching up/down (on the non-clutched train - depending on current rpm trend). Which requires some energy for switching as well as for accelerating/slowing the non-clutched drive train.
What might have some reduction effect: early switching into a higher gear effects in an overall lower figure for the rpms, which may help to reduce fuel consumption (compared to late switching).
Engine "behavior"? Depends on the system you want to compare it to. And on the DCT "drive profile" (early vs. late switching).
There will be improvement in engine efficiency but it will be very miniscule. DCT will be used in Automated Manual Transmission (AMT), in the absence of DCT the engine RPM will be increased / reduced by the ECU / ECM to match the Gearbox input shaft RPM.
In DCT The same effect will be done by the dual clutch.
So if the comparison is between AMT Single clutch and dual clutch it will be miniscule or negligible.
If the comparison is between Manual transmission & AMT Dual clutch means the efficiency will be more (say 5%). Reason is optimal engagement of gears, no over revving of engines. it is designed to maintain the engine RPM at its max efficiency curve / bandwidth,