  We use recent extensions of the Borsuk--Ulam theorem for Stiefel manifolds to
generalize the ham sandwich theorem to mass assignments. A $k$-dimensional mass
assignment continuously imposes a measure on each $k$-dimensional affine
subspace of $\mathbb{R}^d$. Given a finite collection of mass assignments of
different dimensions, one may ask if there is some sequence of affine subspaces
$S_{k-1} \subset S_k \subset \ldots \subset S_{d-1} \subset \mathbb{R}^d$ such
that $S_i$ bisects all the mass assignments on $S_{i+1}$ for every $i$. We show
it is possible to do so whenever the number of mass assignments of dimensions
$(k,\ldots,d)$ is a permutation of $(k,\ldots,d)$. We extend previous work on
mass assignments and the central transversal theorem. We also study the problem
of halving several families of $(d-k)$-dimensional affine spaces of
$\mathbb{R}^d$ using a $(k-1)$-dimensional affine subspace contained in some
translate of a fixed $k$-dimensional affine space. For $k=d-1$, there results
can be interpreted as dynamic ham sandwich theorems for families of moving
points.
