  For initial data $f$ in a subcritical Lorentz space $L^{p,q}(\mathbb{R}^{n})
\hookrightarrow \dot B^{-\frac np}_{\infty,\infty}(\mathbb{R}^n)$
($n<p<\infty$, $1\leq q \leq \infty$), we prove results which imply in
particular that a local in time mild Navier-Stokes solution cannot become
unbounded in the $L^{p,q}(\mathbb{R}^{n})$-norm before it becomes unbounded in
the norm of the larger subcritical Besov space $\dot B^{-\frac
np}_{\infty,\infty}(\mathbb{R}^n)$. In view of the known local theory in such
large Besov spaces, this can be thought of as a `propagation of regularity'
type of result; here, we provide a self-contained local theory (including
scaling-appropriate `blow-up estimates', similar to those established by J.
Leray in the Lebesgue setting) in the Lorentz setting, along with uniqueness
results which imply such propagation of regularity. Our existence results are
based on the method of T. Kato (1984) with Lorentz spaces replacing Lebesgue
spaces throughout, and are given without any reference to the Besov framework.
The uniqueness results are similarly self-contained, extending certain Lebesgue
space methods of Fabes-Jones-Riviere (1972) to the Lorentz setting. We also
establish certain continuity properties of the solutions which are constructed.
(A more detailed abstract is provided in the article itself.)
