We report the results of muon-spin spectroscopy ($mu^+$SR) measurements on the staggered molecular spin chain [pym-Cu(NO$_3$)$_2$(H$_2$O)$_2$] (pym = pyrimidine), a material previously described using sine-Gordon field theory. Zero-field $mu^+$SR reveals a long range magnetically-ordered ground state below a transition temperature $T_mathrm{N}=0.22(1)$ K. Using longitudinal-field (LF) $mu^+$SR we investigate the dynamic response in applied magnetic fields $0< B < 500$ mT and find evidence for ballistic spin transport. Our LF $mu^+$SR measurements on the chiral spin chain [Cu(pym)(H$_2$O)$_4$]SiF$_6 cdot $H$_2$O instead demonstrate one-dimensional spin diffusion and the distinct spin transport in these two systems likely reflects differences in their magnetic excitations.
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