If a fraction \(f_{\rm dcdm}\) of the Dark Matter decays into invisible and massless particles (so-called "dark radiation") with the decay rate (or inverse lifetime) \(\Gamma_{\rm dcdm}\), such decay will leave distinctive imprints on cosmological observables. With a full consideration of the Boltzmann hierarchy, we calculate the decay-induced impacts not only on the CMB but also on the redshift distortion and the kinetic Sunyaev-Zel'dovich effect, while providing detailed physical interpretations based on evaluating the evolution of gravitational potential. By using the current cosmological data with a combination of Planck 2015, Baryon Acoustic Oscillation and redshift distortion measurements which can improve the constraints, we update the \(1\sigma\) bound on the fraction of decaying DM from \(f_{\rm dcdm}\lesssim5.26\%\) to \(f_{\rm dcdm}\lesssim1.99\%\) for the short-lived DM (assuming \(\Gamma_{\rm dcdm}/H_0\gtrsim10^4\)). However, no constraints are improved from RSD data (\(f_{\rm dcdm}\lesssim1.03\%\)) for the long-lived DM (i.e., \(\Gamma_{\rm dcdm}/H_0\lesssim10^4\)). We also find the fractional DM decay can only slightly reduce the \(H_0\) and \(\sigma_8\) tensions, which is consistent with other previous works. Furthermore, our calculations show that the kSZ effect in future would provide a further constraining power on the decaying DM.