Proposition: One-dimensional relay

Let $R\ge 1$ be an integer. On $\mathbb{Z}$, with $d(i,j)=a|i-j|$, use the blocks

Swap the endpoint factors of every block at each tick. A factor initially at $0$ is at $tR$ after $t$ ticks, attaining speed $Ra/\tau$. In units $R=a=\tau =1$, the attained speed is $1$.

Proof. At each tick the blocks are disjoint consecutive intervals of diameter $Ra$. At tick $t$, the transported factor is at the left endpoint $(t-1)R$ of $B_{t,0}$. The endpoint swap moves it to $tR$, which is the left endpoint of $B_{t+1,0}$. Induction proves the claim.

For any finite horizon, retain a finite interval containing all blocks used by this relay. Keep complete blocks inside that interval and complete cut boundary fragments with singleton blocks. No infinite tensor-product state is needed.