  We investigate the Drude weight and the related Mazur-Suzuki (MS) bound in a
broad variety of strongly coupled field theories with a gravity dual at finite
temperature and chemical potential. We revisit the derivation of the recently
proposed universal expression for the Drude weight for Einstein-Maxwell-dilaton
(EMd) theories and extend it to the case of theories with multiple massless
gauge fields. We show that the MS bound, which in the context of condensed
matter provides information on the integrability of the theory, is saturated in
these holographic theories including R-charged backgrounds. We then explore the
limits of this universality by studying EMd theories with $U(1)$ spontaneous
symmetry breaking and gravity duals of non-relativistic field theories
including an asymptotically Lifshitz EMd model with two massless gauge fields
and the Einstein-Proca model. In all these cases, the Drude weight, computed
analytically, deviates from the universal result and the MS bound is not
saturated. In general it is not possible to deduce the low temperature
dependence of the Drude weight by simple dimensional analysis. Finally we study
the effect of a weak breaking of translational symmetry by coupling the EMd
action, with and without $U(1)$ spontaneous symmetry breaking, to an axion
field. We show the coherent part of the conductivity in this limit is simply
the product of the MS bound and the scattering time obtained from the leading
quasinormal mode. For asymptotically $AdS$ theories it seems that the MS bound
sets a lower bound on the DC conductivity for a given scattering time.
