  For each complete meet-semilattice $M$, we define an operator
$\mathsf{Pop}_M:M\to M$ by \[\mathsf{Pop}_M(x)=\bigwedge(\{y\in M:y\lessdot
x\}\cup\{x\}).\] When $M$ is the right weak order on a symmetric group,
$\mathsf{Pop}_M$ is the pop-stack-sorting map. We prove some general properties
of these operators, including a theorem that describes how they interact with
certain lattice congruences. We then specialize our attention to the dynamics
of $\mathsf{Pop}_{\text{Tam}(\nu)}$, where $\text{Tam}(\nu)$ is the
$\nu$-Tamari lattice. We determine the maximum size of a forward orbit of
$\mathsf{Pop}_{\text{Tam}(\nu)}$. When $\text{Tam}(\nu)$ is the $n^\text{th}$
$m$-Tamari lattice, this maximum forward orbit size is $m+n-1$; in this case,
we prove that the number of forward orbits of size $m+n-1$ is
\[\frac{1}{n-1}\binom{(m+1)(n-2)+m-1}{n-2}.\] Motivated by the recent
investigation of the pop-stack-sorting map, we define a lattice path
$\mu\in\text{Tam}(\nu)$ to be $t$-$\mathsf{Pop}$-sortable if
$\mathsf{Pop}_{\text{Tam}(\nu)}^t(\mu)=\nu$. We enumerate
$1$-$\mathsf{Pop}$-sortable lattice paths in $\text{Tam}(\nu)$ for arbitrary
$\nu$. We also give a recursive method to generate $2$-$\mathsf{Pop}$-sortable
lattice paths in $\text{Tam}(\nu)$ for arbitrary $\nu$; this allows us to
enumerate $2$-$\mathsf{Pop}$-sortable lattice paths in a large variety of
$\nu$-Tamari lattices that includes the $m$-Tamari lattices.
