Do particles interact with the Higgs field more at higher speed? It is said that interaction with the Higgs field gives rest mass to fundamental particles.
When a particle is moving, is it then correct to say that there is more of the same interaction that gives it rest mass, thus accounting for its full relativistic mass?  
 A: Alan, I prefer the term invariant mass over rest mass and your question provides an example motivating that preference.
The terms in a relativistic Lagrangian for some QFT must be Lorentz scalars; they must be invariant under Lorentz transformations, i.e., they must have the same value in all reference frames.
When the Higgs field has a vacuum expectation value, the Standard Model Lagrangian can be written such that the matter and gauge fields have associated "mass" terms.  These terms are, of course, Lorentz invariant.
So, in this way, it is clear that the interaction results in an invariant mass, one that is the same in all reference frames.
A: No.
Interaction with Higgs field doesn't depend on motion. That's why Rest Mass is not variable.
Relativistic Mass has nothing to do with Higgs mechanism. It varies because every interaction with mass in relativistic physics are affected by modified space-time scale. For better understanding, energy setup of physics takes care of that. $$E=\sqrt {(mc^2)^2+(pc)^2}$$ It says mass is energy, so there's no reason why it can't be changed.
