What is the process that gives mass to free relativitic particles? When a free particle move in space with a known momentum and energy then what is the physical process that gives mass to that free (relativistic) particle?
What is role does the Higgs field in that process, if any?
 A: It should be stated that the mass of any actual thing that you've encountered in life has almost nothing to do with the Higgs.  Relativity says that energy and mass are equivalent.  This means that if you clump massless particles together with $E$ potential energy, and then put those particles into a box, it will appear that that box has mass $E/c^{2}$.  
Neat, but who cares?  Well, it turns out that the strong interaction does exactly this, clumping quarks together in boxes we call protons and neutrons.  Now, quarks DO get a mass from the Higgs mechanism, but this mass is much, much smaller (roughly $1/1000$) than their "relativistic mass."  So, you would say that you and me, or a rock or a baseball has mass not due to the Higgs mechanism, but rather because all of these things are made mostly of bound quarks.
A: I'll focus on the Higgs aspect of this. There is, potentially, a separate issue to your question, in that you may be thinking about "relativistic mass", the energy an object gains as it speeds up. Modern physicists don't use this concept and I'll explain why underneath. 
Ignore the all-too-common answers that tell you the Higgs field slows down particles, as if it were a sea of molasses. This suggests that the Higgs field imparts mass by exerting a drag effect on them, and is simply incorrect.
So how does it work? First, consider what is meant by "mass". Whilst you may be used to thinking of mass as a property that determines how much something weighs, or how hard it is to move (it is this latter aspect that the $\textit{molasses}$ analogy appeals to), Einstein taught us another way to think of it (the way modern physicists use the word): as "rest energy". 
Energy is a property of a particle that can come either from moving ("kinetic energy") or from its position in a force field ("potential energy"). Einstein's special relativity revealed that some particles also have another form; an intrinsic amount of energy that is always present, regarldess of where the particle is or how fast it is moving. It is this $\textit{intrinsic}$ or $\textit{rest}$ energy that we call mass. (Hence "relativistic mass" is simply the energy gained by speeding up; it is, in other words, just kinetic energy. The word and concept "mass" is best reserved for "rest mass").
We could, if we like, declare that certain particles are born with intrinsic rest energy and leave it there, but for technical reasons this is incompatible with what we know about particle physics, so we are forced to conclude that, intrinsically, all particles are massless, and this rest energy must be given to some of them by something called the Higgs field:
Consider an electric or magnetic field, which emanates from a charged object. Other particles which have an electric charge (or are magnetised) will gain potential energy when they are placed in those respective fields (the amount of energy will depend both on the charge of the particle and the strength of the field). The Higgs field is like an electric field, with one crucial difference: whereas an electric field diminishes in strength the further you get from its source, the Higgs field is constant (and non-zero) throughout space and time. Thus a particle with "Higgs charge" will have the same, constant amount of "Higgs potential energy" wherever it is. This potential energy is, in other words, intrinsic energy, or mass. Larger Higgs-charge leads to larger intrinsic energy = larger mass.
There is, of course, much more to be said on the subject, but that's the essence of how the Higgs field works.
A: As stated in the Lee Smolin’s new book, Time Reborn, the Standard Model (SM), which is based on quantum field theory, is “just an effective theory”. It works well when you consider particles as point-like objects colliding with each other.
As for the "free" particles, SM provides speculative picture of massless objects that do not propagate at the speed of light only due to coupling with the Higgs particles that "slow down" those massless objects. Those Higgs particles that couple with other particles are considered virtual, I guess.
According to the SM, in this process the Higgs particle behaves very selectively with respect to other particles: for instance, it interacts stronger with muons and weaker with electrons for some unknown reason. In absence of Higgs particle, the fields of muon and electron in the SM are indistinguishable. Higgs particle plays the role of the "marker": depending of the "choice of the coupling strength" made by Higgs, massless fermions become electrons, muons, tau etc.
The SM does not explain why:


*

*there are only several types of allowed masses (depending on the strength of coupling with Higgs particle)

*the values of the masses (or coupling constants) are as they are

*the mass of the Higgs particle itself has the value of 126 Gev (as measured by LHC)


I think that this picture is not satisfactory. I agree with Smolin that SM is just an approximation. The SM cannot be used to explain the origin of mass. It is like Kepler law that can be used for identifying the orbit of Earth around the Sun, but cannot be used for identifying, say, the velocity of palm tree growing on equator of the Earth. If one will use Kepler law for identifying the velocity of palm-tree on equator, he of she will find that it has to be 8 km/sec with respect to the center of the Earth. Of course, this is not the case. This is because the physical model based on gravity only (like Keppler law) does not work well for the palm-trees on Earth.
My personal opinion is that mass has electromagnetic origin. See, e.g., this link.
A: General relativity describes gravity (force) and the large-scale structures. Quantum field theory, mainly used in standard model of particle physics, describes small-scale structures and includes 3 forces: weak, strong and electromagnetic. These two theories are not unified. 
According to particle physics, particles gain mass by interacting with Higgs bosons. In general relativity the gain of relativistic mass is due to the object's speed (limited by the speed of light). Classically the speed is not limited so as the object accelerates, its velocity increases and we measure an increase in kinetic energy. In GE the object is 'trying' to accelerate but is limited by the speed of light. In this case instead of measuring increase in kinetic energy we measure an increase in mass because there is no other way of increasing object's energy (kinetic energy approaches maximum value and becomes constant). This is due to conservation of energy.
