What caused the universe to expand in the epoch right after inflation? I hope this exact question is not a duplicate. I have looked into some high rated answers/questions, but found no satisfying/understandable answer. This is my current line of throught:

*

*In the inflation model, a strong expansion of the universe is assumed during the period, after which we start counting time today. This inflation has a yet unknown cause and ends spontaneously.


*Then a slower expansion happens which decelerates due to the energy content of the universe.


*Eventually (due to the cosmological constant), the universe again accelerates to expand.
I do not understand (2). With inflation gone and dark energy not yet strong enough, what fuels this expansion?
EDIT:
It seems to boil down to two ingredients:
a) an initial expansion rate that is given by the termination properties of inflation via an athropic principle approach.
b) an "inertia" of the spacetime expansion, which is necessary to allow the inital value to carry forward into future several billion years.
I want to re-phrase my question: What causes spacetime to have "inertia", i.e. why does its expansion not react instantly on its ingredients?
EDIT 2:
Koschi made me aware of the Friedmann eqs. which describe what I was looking for. The simple picture drawn by popular news outlets that I knew, does not capture some aspects of these eqs.
 A: I think your question concerns what is called the problem of a "graceful exit" at the end of inflation, which is to say, how the conditions before the end of an inflationary episode can lead to the observed conditions after inflation---that is, a highly uniform universe with a very regular expansion. As far as I know it has proved to be very difficult to argue convincingly that inflation leads to a graceful exit; the best we can do (I think---this is just based on general reading) is to place very special conditions on the hypothesized inflationary process.
A: Spacetime doesn't have inertia. The matter in it does, and the expansion is just the expansion of that matter. See this answer.
If you can arrange for reheating to produce a homogeneous, isotropic, expanding matter distribution (the graceful-exit problem mentioned in Andrew Steane's answer), then the continued expansion over billions of years is just (the general-relativistic version of) Newton's first law. Nothing needs to drive it.
A: It is expected that the inflaton decayed into the Standard Model particles, directly, or indirectly, ending inflation. Therefore, after inflation the expansion is assumed to "be fueled" (i.e. driven) by the energy in radiation, meaning actual photons, but also all other particles of the standard model, since they had such high energy that all of them were relativistic, meaning their kinetic energy was much higher than their mass (in natural units). At temperatures higher than the so-called electroweak scale, also it is assumed that the Higgs mechanism was not yet at play, rendering all the particles massless, and relativistic, anyway. After the universe cooled down further, most particles became non-relativistic, and photons have lost a lot of energy because of redshift, so there was a certain time that we expect the universe to be matter dominated... but still, in both eras, there was the right amount of energy to keep the expansion going.
Actually, how radiation and matter can fuel the expansion is a basic discussion in every cosmology textbook and lecture script, so I highly recommend looking into one of these!
A: The present cosmology model is called the "Big Bang" because its first manifestation as a model  was a "Bang" of energy, an explosion of what is now the total energy of the universe,  from a singularity that was allowed to be  within general relativity. It used to be a nice < shape with time starting from the original singularity. It has evolved now to this complicated model

Continued observations of the cosmic microwave radiation, the need of quantum mechanics when distances become very small , that there exists still expansion of the universe, not just kinematic explosion limits, have developed the Big Bang current version above.
Quantum mechanics is used to spread the effects of the singularity over time 0, because a basic tenet of mainstream physics is that the underlying nature of everything should be quantum mechanical, so the singularity becomes fuzzy.
Then the uniformity of CMB necessitated the inflation period, where the expansion was rapid homogenizing the energy everywhere ,and then the particles appeared and slowly as the universe expanded according to the original Bang we come to the observable universe, which is observed to  be still expanding, not just following an initial explosion.
You ask:

What caused the universe to expand in the epoch right after inflation?

What exploded was a lot of mass/ energy , and the explosion   was  fueled in the original Big Bang at time around 0. All explosion have an original energy imparted to the exploding masses, which then continue following the kinematics. In the case of the Big Bang the kinematics are those of general relativity.
With the need for inflation, the assumption is that the original explosion input energy has expanded in time up to the end of inflation, in the plot above. Then it was assumed the original Big Bang explosive impetus took over with a constant expansion.
Except , it has been observed that there is an accelerated   expansion, as seen in the plot . This  is modeled as dark energy, and how this happens is still a matter of on going theoretical and observational research.
