Counting Downsteps in $k_t$Dyck paths

$s_{n, t, r}$ ... total number of downsteps between $r$th and $(r+1)$th
upstep in all $k_t$Dyck paths of length $(k+1)n$ (= $n$ upsteps)
\[ s_{2, 1, 0} = {\color{blue}3}, \quad s_{2, 1, 1} = {\color{lightblue}9},\quad s_{2, 1, 2} = 16 \]
Downsteps before the first upstep
Theorem (Asinowski–H.–Selkirk, 2020+)
The total number of downsteps before the first upstep in all $k_t$Dyck
paths of length $(k+1)n$ is
\[ s_{n, t, 0} = \sum_{j=0}^{t1} C_{n, j}. \]
Proof idea:
 Mark $j$th downstep at start of path
 Cyclic shift: move first $j$ steps to end $\rightsquigarrow$ $k_{tj}$Dyck path
Downsteps between 1st and 2nd upstep in $k_0$Dyck paths
Theorem (Asinowski–H.–Selkirk, 2020+)
The total number of downsteps between the first and second upstep in
all $k_0$Dyck paths of length $(k+1)n$ is
\[ s_{n, 0, 1} = \sum_{j=0}^{k1} C_{n1, j} = \frac{k}{n} \binom{(k+1)(n1)}{n1}. \]
Downsteps everywhere else
Theorem (Asinowski–H.–Selkirk, 2020+)
For all $1\leq r\leq n$, the number of downsteps between the $r$th and
$(r+1)$th upsteps in all $k_t$Dyck paths of length $(k+1)n$ satisfies
\[ s_{n, t, r} = s_{n, t, r1} + C_{r, t} (s_{nr+1, 0, 1}  t[r = n]). \]
Some intuition:
 $r$marked paths: one marked downstep between $r$th and $(r+1)$th upstep
 $s_{n,t,r}$ counts $r$marked $k_t$Dyck paths