A combinatorial bijection on $k$-noncrossing partitions

For any integer $k\ge 2$, we prove combinatorially the following Euler (binomial) transformation identity $${\text{NC}}_{n+1}^{(k)}(t)=t\sum _{i=0}^n\left(\genfrac{}{}{0px}{}{n}{i}\right){\text{NW}}_i^{(k)}(t),$$ where ${\text{NC}}_m^{(k)}(t)$ (resp.~${\text{NW}}_m^{(k)}(t)$) is the sum of weights, $t^{\text{number of blocks}}$, of partitions of $\{1,\dots ,m\}$ without $k$-crossings (resp.~enhanced $k$-crossings). The special $k=2$ and $t=1$ case, asserting the Euler transformation of Motzkin numbers are Catalan numbers, was discovered by Donaghey 1977. The result for $k=3$ and $t=1$, arising naturally in a recent study of pattern avoidance in ascent sequences and inversion sequences, was proved only analytically.