I should preface that I don’t know if this fits within some established theory or not, so the terminology I use may not be canonical.
Terms and definitions:
A Piece $P$ is an $n\times m$ matrix.
A Move $M$ is a $m\times 1$ with integral coefficients.
A Lattice $L$ is the span of the columns of $P$, or $\{PM:M\in \mathbb{Z}^m\}$
The Distance $D$ of a vector $V$ in $L$ is $\min(\{|M|:PM=v\})$ Where $|M|$ is the taxicab metric on $M$
A Sink $S$ in $L$ is a vector $v$ such that $D(V+PM)<D(V)+D(PM)$ $\forall M \neq 0$
Motivation:
If all the vectors in $P$ are integral linearly independent, then L has no sinks because it is isomorphic to $\mathbb{Z}^{m}$ with the taxicab metric.
If $P$ is the set of Knight’s moves, as in chess, then the only sinks are at (2,2), (2,-2), (-2,2), and (-2,-2)
If there is some $K\in \mathbb{Z}$ such that $\forall M$ such that $|M|=K$, $PM$ is not a sink, then $PM:|M|>K$ is not a sink.
The only issue is that there may not be such a $K$. If there is such a case with infinitely many sinks, what is the smallest value of M where this is the case?