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You can't think of the Higgs as something that hitches a ride on other particles to give them mass. The mass of one real Higgs is expected to be hundreds of times greater than the mass of, say, a proton or neutron, let alone quarks and electrons, so it's not as though the mass of a particle come from having a bunch of Higgs bosons as constituent particles. I'm quite sure of that...
Now my hand-waving, shaky understanding of the way the Higgs confers mass centers on the notion of "virtual" particles. One way of looking at the basic idea is that the vacuum is constantly "foaming" with virtual particles. There is not enough energy available to create "real" particles, but there is also always some uncertainty in any energy which means there's a corresponding uncertainty in the number of particles present. (A more down-to-earth example is the photon... you've probably heard that photons are the force mediating particle for electromagnetic interactions, but its not as if there's a stream of real photons flowing from every charged particle. If you want to use quantum field theory to model this you wind up with expressions that look like expressions one would write for a real photon - but they do not conserve energy!)
This is true of any field (you should think of these "particles" not as little balls flying around through space but as something more akin to a pulse-shaped traveling wave on a string). So my way of making sense of the Higgs as generator of mass is that mass arises because of the interaction of a given particle with the surrounding sea of virtual Higgs bosons. Inertia arises because (for some reason) these Higgs interactions conspire to maintain any state of motion for the massive, real particles.
There's probably something wrong with this picture... the main part that bothers me is that it seems there might be a preferred reference frame in which one might say the Higgs bosons are "at rest." Maybe if I knew more field theory I'd see why this isn't what the theory implies. At the same time, there's still no fully-developed quantum theory that's completely consistent with general relativity, which I think leaves a big window for surprises.
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