diff --git a/schemerz/src/lib.rs b/schemerz/src/lib.rs
index c866742..787e1f5 100644
--- a/schemerz/src/lib.rs
+++ b/schemerz/src/lib.rs
@@ -16,13 +16,16 @@ use std::rc::Rc;
 use std::sync::Arc;
 
 use daggy::petgraph::EdgeDirection;
-use daggy::Dag;
+use daggy::{Dag, Walker};
 use indexmap::IndexSet;
 use log::{debug, info};
 use thiserror::Error;
 
+use crate::traversal::DfsPostOrderDirectional;
+
 #[macro_use]
 pub mod testing;
+mod traversal;
 
 /// Metadata for defining the identity and dependence relations of migrations.
 /// Specific adapters require additional traits for actual application and
@@ -318,20 +321,20 @@ where
                     return Err(DependencyError::UnknownId(id));
                 }
             }
-            // This will eventually yield all migrations, so could be optimized.
             None => to_visit.extend(self.dependencies.graph().externals(dir.opposite())),
         }
 
-        let mut target_ids = IndexSet::new();
+        let mut target_set = IndexSet::new();
 
-        while let Some(idx) = to_visit.pop() {
-            if !target_ids.contains(&idx) {
-                target_ids.insert(idx);
-                to_visit.extend(self.dependencies.graph().neighbors_directed(idx, dir));
+        for idx in to_visit {
+            if !target_set.contains(&idx) {
+                let walker = DfsPostOrderDirectional::new(dir, &self.dependencies, idx);
+                let nodes: Vec<daggy::NodeIndex> = walker.iter(&self.dependencies).collect();
+                target_set.extend(nodes.iter());
             }
         }
 
-        Ok(target_ids)
+        Ok(target_set)
     }
 
     /// Apply migrations as necessary to so that the specified migration is
@@ -355,7 +358,7 @@ where
         // TODO: This is assuming the applied_migrations state is consistent
         // with the dependency graph.
         let applied_migrations = self.adapter.applied_migrations()?;
-        for idx in target_idxs.into_iter() {
+        for idx in target_idxs {
             let migration = &self.dependencies[idx];
             let id = migration.id();
             if applied_migrations.contains(&id) {
@@ -407,7 +410,7 @@ where
         }
 
         let applied_migrations = self.adapter.applied_migrations()?;
-        for idx in target_idxs.into_iter() {
+        for idx in target_idxs {
             let migration = &self.dependencies[idx];
             let id = migration.id();
             if !applied_migrations.contains(&id) {
diff --git a/schemerz/src/traversal.rs b/schemerz/src/traversal.rs
new file mode 100644
index 0000000..7c5858f
--- /dev/null
+++ b/schemerz/src/traversal.rs
@@ -0,0 +1,73 @@
+use daggy::{
+    petgraph::{
+        visit::{GraphRef, IntoNeighborsDirected, VisitMap, Visitable},
+        Direction,
+    },
+    Walker,
+};
+
+/// Just like `daggy::DfsPostOrder` but can traverse in either direction
+#[derive(Clone, Debug)]
+pub struct DfsPostOrderDirectional<N, VM> {
+    direction: Direction,
+    /// The stack of nodes to visit
+    stack: Vec<N>,
+    /// The map of discovered nodes
+    discovered: VM,
+    /// The map of finished nodes
+    finished: VM,
+}
+
+impl<N, VM> DfsPostOrderDirectional<N, VM>
+where
+    N: Copy + PartialEq,
+    VM: VisitMap<N>,
+{
+    /// Create a new `DfsPostOrderDirectional` using the graph's visitor map, and put
+    /// `start` in the stack of nodes to visit.
+    pub fn new<G>(direction: Direction, graph: G, start: N) -> Self
+    where
+        G: GraphRef + Visitable<NodeId = N, Map = VM>,
+    {
+        DfsPostOrderDirectional {
+            direction,
+            stack: vec![start],
+            discovered: graph.visit_map(),
+            finished: graph.visit_map(),
+        }
+    }
+
+    /// Return the next node in the traversal, or `None` if the traversal is done.
+    pub fn next<G>(&mut self, graph: G) -> Option<N>
+    where
+        G: IntoNeighborsDirected<NodeId = N>,
+    {
+        while let Some(&nx) = self.stack.last() {
+            if self.discovered.visit(nx) {
+                // First time visiting `nx`: Push neighbors, don't pop `nx`
+                for succ in graph.neighbors_directed(nx, self.direction) {
+                    if !self.discovered.is_visited(&succ) {
+                        self.stack.push(succ);
+                    }
+                }
+            } else {
+                self.stack.pop();
+                if self.finished.visit(nx) {
+                    // Second time: All reachable nodes must have been finished
+                    return Some(nx);
+                }
+            }
+        }
+        None
+    }
+}
+
+impl<G> Walker<G> for DfsPostOrderDirectional<G::NodeId, G::Map>
+where
+    G: IntoNeighborsDirected + Visitable,
+{
+    type Item = G::NodeId;
+    fn walk_next(&mut self, context: G) -> Option<Self::Item> {
+        self.next(context)
+    }
+}