In this paper, we study avoshifts and unishifts on $\mathbb{Z}^d$. Avoshifts are subshifts where for each convex set $C$, and each vector $v$ such that $C \cup \{\vec v\}$ is also convex, the set of valid extensions of globally valid patterns on $C$ to ones on $C \cup \{v\}$ is determined by a bounded subpattern of $C$. Unishifts are the subshifts where for such $C, \vec v$, every $C$-pattern has the same number of $\vec v$-extensions. Cellwise quasigroup shifts (including group shifts) and TEP subshifts are examples of unishifts, while unishifts and subshifts with topological strong spatial mixing are examples of avoshifts. We prove that every avoshift is the spacetime subshift of a nondeterministic cellular automaton on an avoshift of lower dimension up to a linear transformation and a convex blocking. From this, we deduce that all avoshifts contain periodic points, and that unishifts have dense periodic points and admit equal entropy full shift factors.