What exactly causes this kind of damage? Is this just the overheating, or maybe a chemical reaction darkening the filter?
UV light is a form of ionizing radiation. Meaning, it can break chemical bonds and/or liberate electrons from atoms/molecules. Generally, the damage is done by breaking down the substance (i.e., breaking chemical bonds) used in the filter, which produces new "impurities" that have different transmission coefficients in a given energy/frequency range. A similar effect occurs on the conductive glasses used to cover solar panels on spacecraft, where over time they become effectively black due to UV damage. There are some notes on this in the 2015 Senior Review (PDF) proposal for the Wind spacecraft (e.g., see Figure 15 in review showing the degradation of the solar panel current output). The change in slope in Figure 15 is due to the weaker solar cycle (i.e., lower high energy photon fluxes). A more explicit set of results discussion solar cell damage on spacecraft can be found in this NASA Technical Report Study (PDF).
Are there any numbers/formulas available to evaluate the impact of solar radiation on the filter? I did not manage to find anything more specific than "EM waves can damage filters", and I really need to find out what the lifespan of my filter will be and how fast its properties will evolve.
You could do something like determine the work function and/or bond energy of each atom/molecule in the filter substance. Then using known average UV fluxes/intensities, you could estimate how long a given volume of material would last. You can look up ionizing energy tables where the values are given in useful units of kJ/mol, which you can easily convert if you have a known volume of a substance. Figure out the molecular weight of the filter substance, then determine the total number of moles in a given volume, then you can determine the total deposited energy necessary to destroy the entire filter (as a sort of longest duration upper limit, where the true value will actually be shorter).
There is a very detailed explanation of effects and analytical approximations for ionizing radiation damage to solar cells from the Solar Cell Radiation Handbook. The details are mostly about the physical damage to solar cells from high energy electron, proton, and neutron radiation which alters the semi-conductor structure.