  We present a systematic approach to regularity theory of the
multi-dimensional Euler alignment systems with topological diffusion introduced
in \cite{STtopo}. While these systems exhibit flocking behavior emerging from
purely local communication, bearing direct relevance to empirical field
studies, global and even local well-posedness has proved to be a major
challenge in multi-dimensional settings due to the presence of topological
effects. In this paper we reveal two important classes of global smooth
solutions -- parallel shear flocks with incompressible velocity and stationary
density profile, and nearly aligned flocks with close to constant velocity
field but arbitrary density distribution. Existence of such classes is
established via an efficient continuation criterion requiring control only on
the Lipschitz norm of state quantities, which makes it accessible to the
applications of fractional parabolic theory. The criterion presents a major
improvement over the existing result of \cite{RS2020}, and is proved with the
use of quartic paraproduct estimates.
