  Over a field of characteristic $0$, we prove that for each $r \geq 0$ there
exists a constant $C(r)$ so that the prime ideal of the $r$th secant variety of
any Pl\"ucker-embedded Grassmannian ${\bf Gr}(d,n)$ is generated by polynomials
of degree at most $C(r)$, where $C(r)$ is independent of $d$ and $n$. This
bounded generation ultimately reduces to proving a poset is noetherian, we
develop a new method to do this. We then translate the structure we develop to
the language of functor categories to prove the $i$th syzygy module of the
coordinate ring of the $r$th secant variety of any Pl\"ucker-embedded
Grassmannian ${\bf Gr}(d,n)$ is concentrated in degrees bounded by a constant
$C(i,r)$, which is again independent of $d$ and $n$.
