  We study thermodynamic and transport observables of quantum critical states
that arise in the infra-red limit of holographic renormalisation group flows.
Although these observables are expected to exhibit quantum critical scaling,
there are a number of cases in which their frequency and temperature
dependences are in apparent contradiction with scaling theories. We study two
different classes of examples, and show in both cases that the apparent
breakdown of scaling is a consequence of the dependence of observables on an
irrelevant deformation of the quantum critical state. By assigning scaling
dimensions to the near-horizon observables, we formulate improved scaling
theories that are completely consistent with all explicit holographic results
once the dependence on the dangerously irrelevant coupling is properly
accounted for. In addition to governing thermodynamic and transport phenomena
in these states, we show that the dangerously irrelevant coupling also controls
late-time equilibration, which occurs at a rate parametrically slower than the
temperature $1/\tau_{eq}\ll T$. At very late times, transport is
diffusion-dominated, with a diffusivity that can be written simply in terms of
$\tau_{eq}$ and the butterfly velocity, $D\sim v_B^2\tau_{eq}$. We conjecture
that in such cases there exists a long-lived, propagating collective mode with
velocity $v_s$, and in this case the relation $D=v_s^2\tau_{eq}$ holds exactly
in the limit $\tau_{eq} T\gg1$.
