Aims. We present here a first comparative study between the observed damping of numerous fast kink oscillations and the theoretical model of their damping due to the cooling of coronal loops. The theory of damping of kink oscillations due to radiation of the solar plasma with a temporally varying background is applied here to all known cases of coronal kink oscillations. Methods. A recent dynamic model of cooling coronal loops predicts that transverse oscillations of such loops could be significantly damped due to the radiative cooling process (Morton & Erdélyi 2009, ApJ, 707, 750). The cooling of the loop plasma also has the consequence that the kink oscillation has a time-dependent frequency. The theory is applied to a relatively large number of known and reported examples of TRACE observations of damped kink oscillations. Results. We find that, for cooling timescales that are typical of EUV loops (500-2000 s), the observed damping of the transversal (i.e. kink) oscillations can be accounted for almost entirely by the cooling process in half of the examples. No other dissipative mechanism(s) seems to be needed to model the damping. In the remaining other examples, the cooling process does not appear to be able to account fully for the observed damping, though could still have a significant influence on the damping. In these cases another mechanism(s), e.g. resonant absorption, may be additionally required to account for the complete decay of oscillations. Also, we show that because of the dynamic nature of the background plasma, allowing for a time-dependent frequency provides a better fit profile for the data points of observations than a fit profile with a constant frequency, opening novel avenues for solar magneto-seismology.