We present measurements of the evolving extinction cross sections of individual aerosol particles (spanning 700-2500 nm in radius) during the evaporation of volatile components or hygroscopic growth using a combination of a single particle trap formed from a Bessel light beam and Cavity ring-down spectroscopy. For single component organic aerosol droplets of 1,2,6-hexanetriol, polyethylene glycol 400, and glycerol, the slow evaporation of the organic component (over time scales of 1000 to 101300 s) leads to a time-varying size and extinction cross section that can be used to estimate the refractive index of the droplet. Measurements on binary aqueous inorganic aerosol droplets containing one of the inorganic solutes ammonium bisulfate, ammonium sulfate, sodium nitrate, or sodium chloride (over time scales of 1000 to 15 000 s) under conditions of changing relative humidity show that extinction cross-section measurements are consistent with expectations from accepted models for the variation in droplet refractive index with hygroscopic growth. In addition, we use these systems to establish an experimental protocol for future single particle extinction measurements. The advantages of mapping out the evolving light extinction cross-section of an individual particle over extended time frames accompanied by hygroscopic cycling or component evaporation are discussed.