We study a phenomenological class of models where dark matter converts to dark radiation in the low redshift epoch. This class of models, dubbed DMDR, characterizes the evolution of comoving dark-matter density with two extra parameters, and may be able to help alleviate the observed discrepancies between early and late-time probes of the Universe. We investigate how the conversion affects key cosmological observables such as the cosmic microwave background (CMB) temperature and matter power spectra. Combining 3x2pt data from Year 1 of the Dark Energy Survey, Planck-2018 CMB temperature and polarization data, supernovae (SN) Type Ia data from Pantheon, and baryon acoustic oscillation (BAO) data from BOSS DR12, MGS and 6dFGS, we place new constraints on the amount of dark matter that has converted to dark radiation and the rate of this conversion. The fraction of the dark matter that has converted since the beginning of the Universe in units of the current amount of dark matter, zeta, is constrained at 68% confidence level to be <0.32 for DES-Y1 3x2pt data, < 0.030 for CMB + SN + BAO data, and <0.037 for the combined dataset. The probability that the DES and CMB+SN+BAO datasets are concordant increases from 4% for the Lambda CDM model to 8% (less tension) for DMDR. The tension in S-8 = sigma(8)root Omega(m)/0.3 between DES-Y1 3x2pt and CMB + SN + BAO is slightly reduced from 2.3 sigma to 1.9 sigma. We find no reduction in the Hubble tension when the combined data is compared to distance-ladder measurements in the DMDR model. The maximum-posterior goodness-of-fit statistics of DMDR and Lambda CDM model are comparable, indicating no preference for the DMDR cosmology over Lambda CDM.