While emission-selected galaxy surveys are biased towards the most luminous part of the galaxy population, absorption selection is a potentially unbiased galaxy selection technique with respect to luminosity. However, the physical properties of absorption-selected galaxies are not well characterized. Here, we study the excitation conditions in the interstellar medium (ISM) in damped Ly alpha (DLA) absorption-selected galaxies. We present a study of the CO spectral-line energy distribution (SLED) in four high-metallicity absorption-selected galaxies with previously reported CO detections at intermediate (z similar to 0.7) and high (z similar to 2) redshifts. We find further evidence for a wide variety of ISM conditions in these galaxies. Two out of the four galaxies show CO SLEDs consistent with that of the Milky Way inner disc. Interestingly, one of these galaxies is at z similar to 2 and has a CO SLED below that of main-sequence galaxies at similar redshifts. The other two galaxies at z > 2 show more excited ISM conditions, with one of them showing thermal excitation of the mid-J (J = 3, 4) levels, similar to that seen in two massive main-sequence galaxies at these redshifts. Overall, we find that absorption selection traces a diverse population of galaxies.