  We investigate the effects of a magnetic field on the dynamics of
rotationally inelastic collisions of open-shell molecules ($^2\Sigma$,
$^3\Sigma$, and $^2\Pi$) with closed-shell atoms. Our treatment makes use of
the Fraunhofer model of matter wave scattering and its recent extension to
collisions in electric [M. Lemeshko and B. Friedrich, J. Chem. Phys.
\textbf{129}, 024301 (2008)] and radiative fields [M. Lemeshko and B.
Friedrich, Int. J. Mass. Spec. in press (2008)]. A magnetic field aligns the
molecule in the space-fixed frame and thereby alters the effective shape of the
diffraction target. This significantly affects the differential and integral
scattering cross sections. We exemplify our treatment by evaluating the
magnetic-field-dependent scattering characteristics of the He -- CaH ($X
^2\Sigma^+$), He -- O$_2$ ($X ^3\Sigma^-$) and He -- OH ($X ^2\Pi_{\Omega}$)
systems at thermal collision energies. Since the cross sections can be obtained
for different orientations of the magnetic field with respect to the relative
velocity vector, the model also offers predictions about the
frontal-versus-lateral steric asymmetry of the collisions. The steric asymmetry
is found to be almost negligible for the He -- OH system, weak for the He --
CaH collisions, and strong for the He -- O$_2$. While odd $\Delta M$
transitions dominate the He -- OH $(J=3/2,f \to J', e/f)$ integral cross
sections in a magnetic field parallel to the relative velocity vector, even
$\Delta M$ transitions prevail in the case of the He -- CaH ($X ^2\Sigma^+$)
and He -- O$_2$ ($X ^3\Sigma^-$) collision systems. For the latter system, the
magnetic field opens inelastic channels that are closed in the absence of the
field. These involve the transitions $N=1, J=0 \to N', J' $ with $J'=N'$.
