  We prove an upper bound on the diffusivity of a general local and translation
invariant quantum Markovian spin system: $D \leq D_0 + \left(\alpha \,
v_\text{LR} \tau + \beta \, \xi \right) v_\text{C}$. Here $v_\text{LR}$ is the
Lieb-Robinson velocity, $v_\text{C}$ is a velocity defined by the current
operator, $\tau$ is the decoherence time, $\xi$ is the range of interactions,
$D_0$ is a microscopically determined diffusivity and $\alpha$ and $\beta$ are
precisely defined dimensionless coefficients. The bound constrains quantum
transport by quantities that can either be obtained from the microscopic
interactions ($D_0, v_\text{LR}, v_\text{C},\xi$) or else determined from
independent local non-transport measurements ($\tau,\alpha,\beta$). We
illustrate the general result with the case of a spin half XXZ chain with
on-site dephasing. Our result generalizes the Lieb-Robinson bound to constrain
the sub-ballistic diffusion of conserved densities in a dissipative setting.
