  We study several different types of BPS flows within minimal $\mathcal{N}=1$,
$D=7$ supergravity with $\textrm{SU}(2)$ gauge group and non-vanishing
topological mass. After reviewing some known domain wall solutions involving
only the metric and the $\mathbb{R}^{+}$ scalar field, we move to considering
more general flows involving a "dyonic" profile for the 3-form gauge potential.
In this context, we consider flows featuring a $\textrm{Mkw}_{3}$ as well as an
$\textrm{AdS}_{3}$ slicing, write down the corresponding flow equations, and
integrate them analytically to obtain many examples of asymptotically
$\mathrm{AdS}_7$ solutions in presence of a running 3-form. Furthermore, we
move to adding the possibility of non-vanishing vector fields, find the new
corresponding flows and integrate them numerically. Finally, we discuss the
eleven-dimensional interpretation of the aforementioned solutions as effective
descriptions of $\mathrm{M2}-\mathrm{M5}$ bound states.
