The broadening of emission lines is not due to something that is happening to each individual star, but rather something that affects the whole population.
As stars in a galaxy get older, their orbits change relative to the orbits of other stars of the same age. Most relevant for the discussion here, the "velocity dispersion" of a stellar population increases with time. Velocity dispersion can be thought of as the standard deviation of the velocity distribution of a stellar population, and in fact is defined as such.
As the velocity dispersion of a stellar population increases with age, stars will be moving at increasingly different velocities relative to each other. The emission lines we measure from these stellar populations thus broaden with time, because the emission lines we see from, e.g., galaxies are not from individual stars but are composite spectra from billions of stars. Because the stars are moving at different velocities relative to the observer, some stars have their spectrum redshifted and other have their spectrum blueshifted relative to what would be the intrinsic spectrum seen from the given stellar population if all the stars were at rest relative each other. The larger the velocity dispersion, the broader the emission lines get as spectra from individual stars become increasingly red and blueshifted.
In short, as velocity dispersion increases, more and more stars are contributing spectra that have emission lines further and further offset from what is the center of the composite line, so the observed emission line (which is a composite of the emission lines of all the stars) broadens.
In practice, velocity dispersions are measured from spectral line widths, since it is impossible to measure velocities of individual stars for all but the nearest galaxies.
As for why velocity dispersion increases with time: stars for the most part see a smooth gravitational potential from the summed gravity of all the stars, gas, dark matter, etc. of a galaxy, but as stars pass "close" (they're still pretty far away) to other stars and gas clouds, they receive small and randomly oriented kicks that cause their orbits to slowly change with time. Mathematically speaking, over time the dynamics of a stellar populations diffuses in phase space. The longer a stellar population has been around, the more kicks each of its stars have received, the greater the diffusion in phase space and the greater the spread in velocities.
There are some other effects that affect line broadening, both on an individual star level and on a population level, but as I understand it these effects (some of which lead to narrowing of the lines, instead of broadening) are sub-dominant relative to the effects of the changing dynamics of the stellar orbits.
I would also add that, for optical/NIR light, over a wavelength range as small as a typical emission line, extinction affects all wavelengths equally and would have no effect on the width of a line.