We present spectrum synthesis fits to 808 K+A galaxies selected from the Sloan Digital Sky Survey (SDSS) and population synthesis of their spectral energy distributions, extending from the far-UV (0.15 mu m) to the mid-IR (22 mu m), based on the results of STARLIGHT code fitting to the SDSS spectra. Our modelling shows that K+A galaxies have undergone a large starburst, involving a median 50 per cent of their present stellar masses, superposed over an older stellar population. The metal abundance of the intermediate-age stars shows that star formation did not take place in pristine gas, but was part of a dramatic increase in the star formation rates for originally gas-rich objects. We find no evidence for ongoing QSO activity in the UV, which is well modelled by the emission of intermediate-age stars. We use K+A galaxies as local counterparts of high-redshift objects to test for the presence of thermally pulsing asymptotic giant branch (TP-AGB) stars in similarly aged populations and find no excess in the infrared due to emission from such stars, arguing that more distant galaxies are indeed old and massive at their redshift. All of our galaxies show significant excesses in the mid-IR compared to the light from their stars. We fit this ad hoc with a 300 K blackbody. Possible sources include TP-AGB stars, obscured young star clusters and hidden AGNs, heating a significant dust component.