Very hard to know if the higgs boson is moving or not our understanding is too early as yet.

Interaction of matter with it gives some mass what we call relativistic mass but it does not create the mass we attribute to gravity. So some otherwise massless particles gain mass by interaction with the higgs but not all.

So again it's not simple the story of the higgs mass depends on what type of object you are talking about.
I believe in "Evil, Bad, Ungodly fantasy science and maths", so I am undoubtedly wrong to you.