Summary: For each 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 \(\mathsf{Pop}_{\operatorname{Tam} (\nu)}\), where \(\operatorname{Tam}(\nu)\) is the \(\nu\)-Tamari lattice. We determine the maximum size of a forward orbit of \(\mathsf{Pop}_{\operatorname{Tam} (\nu)}\). When \(\operatorname{Tam}(\nu)\) 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 \[ \frac{1}{n-1} \begin{pmatrix} (m+1) (n-2) + m-1 \\ n - 2 \end{pmatrix}. \] Motivated by the recent investigation of the pop-stack-sorting map, we define a lattice path \(\mu \in \operatorname{Tam}(\nu)\) to be \(t\)-\(\mathsf{Pop}\)-sortable if \(\mathsf{Pop}_{\operatorname{Tam} (\nu)}^t (\mu) = \nu\). We enumerate \(1\)-\(\mathsf{Pop}\)-sortable lattice paths in \(\operatorname{Tam}(\nu)\) for arbitrary \(\nu\). We also give a recursive method to generate \(2\)-\(\mathsf{Pop}\)-sortable lattice paths in \(\operatorname{Tam}(\nu)\) 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.