First of all it is necessary to emphasize that all the techniques used to solve the continuous-time Boltzmann equation in one variable (plus time)

may be used to solve its discrete-time equivalent

the difficulty depending only on the kernel structure and not on the left-hand side.
The domain W of integration is the appropriate domain given by the problem and is supposed not to have any dependence on the variable x.

• Should there be any such dependence, first of all, move it from the limits of integration into the kernel.

There is not a big difference between the two cases. Integro-differential equations arise mostly in multi-variable problems (discussed later). In single-variable (plus time) equations it is convenient to express any differential operator in integral form and look for the eigenfunctions of the integral operator resulting by the sum of the latter and the former existing integral one.
An example will clarify:

can be rewritten as

where the new kernel is:

From now on therefore, for the single-variable case, only integral equations will be taken into account. A different approach will be used in the multi-variable case.

These are usually referred to as those kernels of the form

No need for any help here; just perform the integral and the equation is almost solved. It is the simplest existing case of Boltzmann equation.

Equations containing kernels of the form

are always solvable without any Feynman Propagator intervention.

These kernels are defined by the fact that they admit as eigenfunctions the powers (with real or sometimes complex exponents) of the variable x; sometimes excluding a limited number of these.

Such equations are solvable and their solvability does not depend on the form of the kernel but only on the mentioned eigenfunctions.
In most cases, however, (as it is in the case of elastic isotropic scattering) the solution takes the form of an integral in the complex plane since it is not expressible in terms of elementary functions.

Among them, those kernels expressible as a sum of products of functions of the two variables,

are called "degenerate" and are solvable equations.

In short, if you know the complete set of eigenfunctions for your kernel the equation can be solved. It often happens (in truth this is the most common case) that on rebuilding the final solution you find integrals that can not be expressed in terms of elementary functions.
This is a consequence of the fact that integral theories are not local: therefore their solutions have properties linked to their integrals.
Known functions are mostly solutions of differential equations which are equations expressing relations between a function and its properties in a fixed point.
New and recurring integrals expressing the properties of new functions must be sought.
Other general integro-differential linear equations who do not fall under this classification are tied to the problem mentioned in the last paragraph.

This is much more complicated than the mono-variable ones.

Here D is a linear differential operator of the form

where the w are given functions of the assigned variables.

Equations ruled by kernels of the form

can be solved as soon as the eigenfunctions of the differential operator can be found. This is a recurring problem also in other cases. Sometimes equations for eigenfunctions can not be solved in explicit form. This means that your degree of knowledge of the equations drops dramatically. Sometimes, however, something can still be done.

These equations are characterized by kernels of the following type

and can be solved as long as the eigenfunctions of the differential operator can be written in explicit form.

As it happens in the single-variable case, equations containing kernels like

can generally be transformed into systems of partial differential equations and then solved (unless there are not the aforementioned problems).

When facing multi-variable cases, many combinations of kernel type arise, such as

No real problem with such kernels has come out till now but as it seems from its structure, the solution might be found with a simple combination of the techniques reported earlier.

Another kind of combination which may well show up is

Same as Mixed Dirac-Delta-B.G.K. Kernels.

Whenever you are facing a different type of equation which does not fit under one of the classes above, do the following, check whether any transformation can cast your equation into one of the classes mentioned above. If you are unlucky, go to the last paragraph.
I have seen several problems in physics and engineering where equations can be successfully transformed into solvable ones.
I report here a solved equation which applies both to neutron transport in a gravity field or to any particle transport problem in the presence of a field of force (e. g. conduction problems).
It can also be found in Kittel's book on solid-state physics (Appendix F).

The equation covering neutron transport in a field of force F

where I have adopted the kernel

and the initial condition

Other kernels and initial conditions can of course be inspected.
In conclusion, once you have found the appropriate treatment of the initial condition, the solution can be given in a form containing several terms.
Two of these are integrals which, so far, resist to the best computer symbolic algebra.

I am working to a general technique to build eigenfunctions of composite operators. If I succeed any equation whose operator can be decomposed in products and sum of known operators will be solvable: clearly this page will be then updated.
The theory covers also time dependence in the kernel and in the differential operator but in practice eigenfunctions always figure in implicit form or involve dramatic integrals: these are hardly expressible in terms of known functions.
I have never focused on such equations nor I have ever tried to solve a particular one.....

.... but your case may be the lucky one.