In order to go into your question I must first say some things about doing physics in general.
In the history of physics we see a progression to unification. A vivid example is the way Kepler's three laws of planetary motion were unified by Newton's laws of motion. Newton's law of universal gravity holds good universally: that is in itself sufficient reason to regard it proven beyond reasonable doubt. At the same time, gravity itself is unexplained
Throughout the history of physics you clearly see that succesful physicists never allowed themselves to be bogged down by insisting on ever deeper explanation. It's a judgement call; Newton did move beyond Kepler's laws, but Newton accepted gravity as is. You pick your battles.
Inertia
As far as I'm aware of, in mainstream physics inertia is regarded as something that must be granted as is, in order to frame any theory of motion. As explained above, probably that is the best possible attitude. If a problem has no prospect of making progress then don't let that bog you down.
The closest analogy for inertia that we have is an electromagnetic phenomenon that is called 'inductance'. A coil with self-induction will oppose change of current strength. When you apply an electromotive force you get a steady rise of current strength. If you have a current going, and you suddenly open the circuit, an electric spark will jump the gap, because the self-induction opposes change of current strength.
Some additional remarks:
In the comment section someone mentioned the Higgs field. I must emphasize: the Higgs field is not an attempt at accounting for inertia.
For instance, in the Standard Model neutrino's are described as having non-zero mass, but that mass is not described as arising through interaction with the Higgs field.
For more information see the 2007 article 'The origin of mass' by Jim Pivarski, particle physicist
Quote from Jim Pivarski:
[...] protons and neutrons are known to be made of quarks, bound by an
incredibly strong force called The Strong Force. Converted into
conventional units, quarks attract each other with forces typically
greater than 15 tons. The potential and kinetic energy of the quark
orbits account for 99% of the mass of protons and neutrons; only the
last 1% is due to the mass of the quarks themselves. This is
relativistic mass in an extreme case — we are made, almost entirely,
out of the attraction of quarks.
About relativistic mass:
Relativistic physics describes that energy that is confined to a region of space has a corresponding inertial mass.
That is why I emphasize that the Higgs field is not an attempt at accounting for inertia.
