  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.
