We study the dynamics of spin currents in the spin-$\frac{1}{2}$ XX ladder at finite temperature. Within linear response theory, we numerically calculate autocorrelation functions for quantum systems larger than what is accessible with exact diagonalization using the concept of dynamical quantum typicality. While the spin Drude weight vanishes exponentially quickly with increasing system size, we show that this model realizes standard diffusive dynamics. Moreover, we unveil the existence of three qualitatively different dependencies of the spin-diffusion coefficient on the rung-coupling strength, resulting from a crossover from exponential to Gaussian dissipation as the rung coupling increases, in agreement with analytical predictions. We further discuss the implications of our results for experiments with cold atomic gases.

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