  An easy-plane spin winding in a quantum spin chain can be treated as a
transport quantity, which propagates along the chain but has a finite lifetime
due to phase slips. In a hydrodynamic formulation for the winding dynamics, the
quantum continuity equation acquires a source term due to the transverse
vorticity flow. The latter reflects the phase slips and generally compromises
the global conservation law. A linear-response formalism for the nonlocal
winding transport then reduces to a Kubo response for the winding flow along
the spin chain, in conjunction with the parasitic vorticity flow transverse to
it. One-dimensional topological hydrodynamics can be recovered when the
vorticity flow is asymptotically small. Starting with a microscopic spin-chain
formulation, we focus on the asymptotic behavior of the winding transport based
on the renormalized sine-Gordon equation, incorporating phase slips as well as
Gilbert damping. A generic electrical device is proposed to manifest this
physics. We thus suggest winding conductivity as a tangible concept that can
characterize low-energy dynamics in a broad class of quantum magnets.
