The work done is the line integral of the sum of the forces acting on the object. $$W=\int\limits_{\ell=0}^{\ell=h}(\vec{F}_{\rm lift}+\vec{F}_g)\cdot d\vec{l} =mgh,$$ where $\vec{F}_g$ is in the opposite direction to $\vec{F}_{\rm lift}$.
The only way to can get the object to move upwards is if the magnitudes $F_{\rm lift} > F_g$.
However, if you want it to stop at $h$, then at some point in the second phase of the motion then the magnitudes $F_{\rm lift} < F_g$.
If you apply a constant lifting force, then of course the body will not be at rest when it reaches $h$. You will have done more work and the extra work will be accounted for by the kinetic energy of the body.