Why does an interaction see the proton instead the quark mass? When an interaction goes deep in the nucleus, should we have some way to interpolate between the proton mass and the quark mass which is going to be the ultimate destination of the interaction? For instance, if the energy scale of the probe is $E$, consider that it sees all the volume around a radius $1/E$ and then set for a mass generically $\propto \rho /E^3$, being the pole quark mass, $3-8  \ MeV$ in the high energy limit and current quark mass, $330\  MeV$ in low energy.
Also, I am not sure of which is the right low-energy limit. I had expected, as I say, to be the quark current mass, but nuclear processes, not only beta decay but also some nuclear scattering, seem to use always the mass of the nucleon $M_N$, even for electroweak corrections (terms such as $\log M_W/M_N$, as Sirlin does) My understanding of it (from reviews as Formaggio-Zeller 2013) is that we are dealing with the infrared case here, but then on the same token they should use the whole nucleus, which they do not.
I guess that generically charge distributions are collected into the form factors. But I am more worried about the mass because it has a role in propagators and also in second-order corrections, such as the logarithmic ones of Sirlin formulae for nucleons.
 A: I would say the approach here is based on scale. I have no references on this, I am exposing my educated guess.
I think there is a relation between the energy transfer and the relevant scale receiving this energy. That is how I would explain the difference between pushing a wood piece, with a rather slow movement, while a skilled martial artist is able to break it with a sudden movement.
This is also evident when observing a fan rotating and trying to shoot a paper ball between two blades: if you shoot at slow speeds, the fan behaves like a wall, impenetrable. This is because their movement is much faster and their average occupation of space is such that the probability of passage is small. Only at high speeds the penetration holes become apparent. 
This in my mind is completely analogous to the collision between any two physical bodies. And the qualitative rule is: the collision is an energy transfer that occurs in a time interval and in a certain volume or area, which completely define the expected result. These time interval and volume area of interest are completely related to the relative momenta of the bodies.
You can throw a baseball to a window, and it won't break as long as the speed is low enough. Because at these speeds the whole window is involved in stopping the ball, the whole window receives the tension of the impact and that tension can distribute throughout the surface in the time scale involved. 
For a faster throw, the ball will break the window, or cause fractures that go all along the surface, long lines of rupture. Here I would say that the energy transfer in the impact was so sudden (a smaller time scale than before) that portions of the window vibrated w.r.t to other in amplitudes which decoupled them. And probably the relevant scale here is represented by the average size of the pieces.
One can see that for even faster speeds, only a hole close to the size of the ball is left of the collision, where energy stayed in the pieces broken. Transfer was here faster than the transmission to other parts of the window, so fast that the other regions didn't notice.
In your answer I see an analog. Although in the quantum world particles cannot be localized, nor their momenta measured with the same precision, something analogous happens. It is well known that wavelength plays an important role in collisions. Neutrons of low energy can scatter on atoms much like balls, and the more their energy approaches the keV range, at which their wavelengths are comparable to nuclear scales, the more they excite the nuclear energy levels and get captured, where nuclear resonances appear in the spectra.
So in the case of electron, only at energies high enough, and wavelengths small enough could it interact with the quarks. This also means that the appropriate timescales became relevant at these energies, since quarks are supposed to have a large momenta distribution inside protons, and at lower speeds the interaction would be similar to the fan blades example.
