My answer refers to the case, that the external field is homogeneous over some area and you move your magnet inside this area. Otherwise see Emilio’s answer.
If the magnet is moving and the charge is stationary then here should be no force on the charged particle...
... is a right statement. Furthermore there is a second surprising case. A homopolar generator (Faraday disc) shows the next behavior:
If the magnetic field is provided by a permanent magnet, the generator works regardless of whether the magnet is fixed to the stator or rotates with the disc.
If you understand how the electron in an external magnetic field gets deflected Lorentz force then you are able to explain both phenomena.
An electron obey a magnetic dipole moment. This intrinsic magnet gets aligned in an external magnetic field. Now, if the electron moves the alignment is accompanied by a deflection due to the gyroscopic effect. Since the deflection is an acceleration the electron radiates. This emission of a photon disalign the tiny magnet again and this cycle repeats until the electron comes to rest in the center of its spiral path.
For your case, the electrons magnetic dipole moment gets aligned, but since it is not moving, no deflection occurs.
For the case the magnet rotates together with the disc, the electron moves in circles and is accelerated, emits photons, gets aligned by the magnetic field again and the electron moves outwards or inwards (depends from the direction of rotation and the direction of the magnetic field) despite the magnetic field is in rest to the disc.