This is a pretty broad question - can be answered many different ways. Need to define some boundary conditions and what are you looking at, ie. Electrostatics, Electromagnetics, constant E field, constant B field, etc. I'm a Physicist, if that matters.
Electrostatics are easy to generalize as they have a very strong parallels to gravity. Once you throw in motion, then you have a time-varying fields. Because these test charges will generate their own E-field and in turn, because they are moving, will generate a B-field as well. The transistion in which electric or magnetic fields change in time is no longer electrostatics, but now you have electromagnetics. Since you mentioned electric field initially and then added magnetic field later - I'll take it as a electromagnetic case and charged particles that are injected, teleported, or leaked into the fields.
You can describe the forces exerted onto the charged particles as Lorentz Force. There will be a force component from the electric field (qE) and from the magnetic field (qv x B ), which together is in the form of F = q(E + v x B ).
Use dimensional analysis (just looking at the units) - you can see that doubling the test charge from a +q to a +2q and assuming that the E and B fields are constant, the net force constant - your velocity for the larger charge would be lower than the velocity of the smaller charge.