The large-scale dynamics of quantum integrable systems is often dominated by ballistic modes due to the existence of stable quasiparticles. We here consider as an archetypical example for such a system the spin-1/2 XXX Heisenberg chain that features magnons and their bound states. An interesting question, which we here investigate numerically, arises with respect to the fate of ballistic modes at finite temperatures in the limit of zero magnetization m=0. At a finite magnetization density m, the spin-autocorrelation function Pi(x, t) (at high temperatures) typically exhibits a trimodal behavior with left- and right-moving quasiparticle modes and a broad center peak with slower dynamics. The broadening of the fastest propagating modes exhibits a subdiffusive t(1/3) scaling at large magnetization densities m -> 1/2, familiar from noninteracting models; it crosses over into a diffusive scaling t(1/2) upon decreasing the magnetization to smaller values. The behavior of the center peak appears to exhibit a crossover from transient superdiffusion to ballistic relaxation at long times. In the limit m -> 0, the weight carried by the propagating peaks tends to zero; the residual dynamics is carried only by the central peak; it is sub-ballistic and characterized by a dynamical exponent z close to the value 3/2 familiar from Kardar-Parisi-Zhang (KPZ) scaling. We confirm that, employing elaborate finite-time extrapolations, that the spatial scaling of the correlator Pi is in excellent agreement with KPZ-type behavior and analyze the corresponding corrections.