Exact target-adapted payload transfer

Choose a straight developed segment from $x_A$ to $x_B$ of length $L>0$, entirely inside $U$, and an integer $n\ge 1$. Install $n+1$ sites along it with spacing $L/n$. Each site carries $A_j\cong {\mathbb{C}}^d$. Prepare $A_1,\dots ,A_n$ in an independent fixed blank $\omega$, and at tick $j$ SWAP $A_{j-1}$ with $A_j$. Set the tick duration to $L/(nc)$.