The standard incidence locus is closed

A point $[v]$ belongs to an $r$-plane with decomposable wedge $w$ exactly when $v\wedge w=0$. Expanding this wedge gives homogeneous bilinear equations in the point and Plucker coordinates.

These equations vanish precisely on ${\mathcal{I}}_{1,r}(V)$ by the annihilator characterization used for the Plucker map. The ambient product is closedly modeled by Segre and Plucker coordinates, so their common zero locus there is closed.

For the containment locus, cover both Grassmannians by their finitely many standard affine charts. On a product of two such charts, the planes $A$ and $B$ have canonical row-frame matrices. The condition $A\subseteq B$ is equivalent to the stacked $(a+b)\times \dim V$ matrix having rank at most $b$, which is cut out by all of its $(b+1)\times (b+1)$ minors. Thus the containment locus is closed on every member of this finite open cover. Closedness is local on an open cover, so it is closed globally. Together with steps 1.1--2.1 this proves both assertions without choosing global basis vectors of the varying plane $A$.