In this paper we consider the energy and momentum transport in (1+1)-dimension conformal field theories (CFTs) that are deformed by an irrelevant operator TT , using the integrability based generalized hydrodynamics, and holography. The two complementary methods allow us to study the energy and momentum transport after the in-homogeneous quench, derive the exact non-equilibrium steady states (NESS) and calculate the Drude weights. Our analysis reveals that all of these quantities satisfy universal formulae regardless of the underlying CFT, thereby generalizing the universal formulae for these quantities in pure CFTs. We also compute the exact momentum diffusion constant using the integrability-based method, and confirm that it agrees with the conformal perturbation. These fundamental physical insights have important consequences for our understanding of the TT-deformed CFTs. First of all, they provide the first check of the TTdeformed AdS3/CFT2 correspondence from the dynamical standpoint. And secondly, we are able to identify a remarkable connection between the TT-deformed CFTs and reversible cellular automata.