Magnetic forces on charges

It has been found experimentally that a magnetic field exerts a force $F$ on a charge which is proportional to the velocity $v$ of the charge. If $v$ is zero, then $F$ is zero.

I assume that you are talking about elementary charges such as electrons. The magnetic field of an electron is a dipole. This is its intrinsic property and is not induced externally. If you bring an electron into a homogeneous external magnetic field (the electrostatic case), both poles of the electron are attracted to the external field with the same force and the electron is not moved. However, the electron is rotated because the dipole of the electron aligns with the external field.

When the electron has kinetic energy and is not moving parallel to an external magnetic field, the magnetic dipole of the electron aligns again and this time the electron radiates (emits a photon) and the recoil deflects the electron laterally. As a result, the electron describes a spiral path until its kinetic energy (in the form of radiation) is emitted.

If the external magnetic field changes dynamically or is inhomogeneous, then the electron with its magnetic dipole is also deflected. All this is included in the equation of the Lorentz force.