Let $F$ be a global field of positive characteristic, and let ${\mathcal D}$ be a maximal order in a central division algebra over $F$ . We give a precise properness criterion for moduli spaces of ${\mathcal D}$ -shtukas with general modifications, which includes the statement that for some division algebras, the space of ordinary ${\mathcal D}$ -shtukas is not proper. The proof is based on a result on the smoothness of a suitable stack of complete homomorphisms of ${\mathcal D}$ -modules of rank $1$