  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 $T\bar{T}$, 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 and
the diffusion constants. 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. As a
sanity check, we also confirm that the exact momentum diffusion constant agrees
with the conformal perturbation. These fundamental physical insights have
important consequences for our understanding of the $T\bar{T}$-deformed CFTs.
First of all, they provide the first check of the $T\bar{T}$-deformed
$\mathrm{AdS}_3$/$\mathrm{CFT}_2$ correspondence from the dynamical standpoint.
And secondly, we are able to identify a remarkable connection between the
$T\bar{T}$-deformed CFTs and reversible cellular automata.
