  We theoretically study transport properties in one-dimensional interacting
quasiperiodic systems at infinite temperature. We compare and contrast the
dynamical transport properties across the many-body localization (MBL)
transition in quasiperiodic and random models. Using exact diagonalization we
compute the optical conductivity $\sigma(\omega)$ and the return probability
$R(\tau)$ and study their average low-frequency and long-time power-law
behavior, respectively. We show that the low-energy transport dynamics is
markedly distinct in both the thermal and MBL phases in quasiperiodic and
random models and find that the diffusive and MBL regimes of the quasiperiodic
model are more robust than those in the random system. Using the distribution
of the DC conductivity, we quantify the contribution of sample-to-sample and
state-to-state fluctuations of $\sigma(\omega)$ across the MBL transition. We
find that the activated dynamical scaling ansatz works poorly in the
quasiperiodic model but holds in the random model with an estimated activation
exponent $\psi\approx 0.9$. We argue that near the MBL transition in
quasiperiodic systems, critical eigenstates give rise to a subdiffusive
crossover regime on finite-size systems.
