-- If the ends of the wire loop are open (not connected to anything) then there's never any current in it, no matter what you do with the magnet.
-- If there's some kind of conductor between the ends of the wire loop, then
-- While you're pushing the magnet, a current is flowing in the wire. The current creates a magnetic field around the wire, and some of the energy you're using to push gets stored in the magnetic field.
-- When you stop pushing and the magnet stops moving, the energy stored in the magnetic field keeps the current flowing in the wire for a short time, and then everything stops.
This all happens so fast, and lasts for such a short time, that you'd never notice it unless you were set up with special equipment to measure it.
