  We measure the transport properties of two-dimensional ultracold Fermi gases
during transverse demagnetization in a magnetic field gradient. Using a
phase-coherent spin-echo sequence, we are able to distinguish bare spin
diffusion from the Leggett-Rice effect, in which demagnetization is slowed by
the precession of spin current around the local magnetization. When the
two-dimensional scattering length is tuned to be comparable to the inverse
Fermi wave vector $k_F^{-1}$, we find that the bare transverse spin diffusivity
reaches a minimum of $1.7(6)\hbar/m$, where $m$ is the bare particle mass. The
rate of demagnetization is also reflected in the growth rate of the s-wave
contact, observed using time-resolved spectroscopy. At unitarity, the contact
rises to $0.28(3) k_F^2$ per particle, measuring the breaking of scaling
symmetry. Our observations support the conjecture that in systems with strong
scattering, the local relaxation rate is bounded from above by $k_B T/\hbar$.
