Depends on what how far you are willing to go. If you have a solenoid with a uniform time-varying magnetic field, then an electric field is induced by Faraday's law. It will be a circular field, and will make an electron at rest spiral. So the electron at rest is affected by a time-varying magnetic field, though indirectly--via an induced electric field.
Without considering induced fields:
An electron will align its spin with a uniform magnetic field, and if the field is nonuniform, it will move along with it. This is because the electron has a magnetic "spin" or dipole moment,
As far as neutrons are concerned, it depends if they have an electric dipole moment(afaict unknown). But, they can be affected in a similar manner by time-varying nonuniform electric fields, since they also have a magnetic moment.