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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 …