  We study the non-equilibrium dynamics of conformal field theory (CFT) in 1+1
dimensions with a smooth position-dependent velocity $v(x)$ explicitly breaking
translation invariance. Such inhomogeneous CFT is argued to effectively
describe 1+1-dimensional quantum many-body systems with certain inhomogeneities
varying on mesoscopic scales. Both heat and charge transport are studied,
where, for concreteness, we suppose that our CFT has a conserved U$(1)$
current. Based on projective unitary representations of diffeomorphisms and
smooth maps in Minkowskian CFT, we obtain a recipe for computing the exact
non-equilibrium dynamics in inhomogeneous CFT when evolving from initial states
defined by smooth inverse-temperature and chemical-potential profiles
$\beta(x)$ and $\mu(x)$. Using this recipe, the following exact analytical
results are obtained: (i) the full time evolution of densities and currents for
heat and charge transport, (ii) correlation functions for components of the
energy-momentum tensor and the U$(1)$ current as well as for any primary field,
and (iii) the thermal and electrical conductivities. The latter are computed by
direct dynamical considerations and alternatively using a Green-Kubo formula.
Both give the same explicit expressions for the conductivities, which reveal
how inhomogeneous dynamics opens up the possibility for diffusion as well as
implies a generalization of the Wiedemann-Franz law to finite times within CFT.
