diff --git a/pooch/processors.py b/pooch/processors.py
index dfeebb38..0e6eff2f 100644
--- a/pooch/processors.py
+++ b/pooch/processors.py
@@ -124,16 +124,30 @@ def __call__(self, fname, action, pooch):
 
         # Get a list of all file names (including subdirectories) in our folder
         # of unzipped files, filtered by the given members list
-        fnames = []
-        for path, _, files in os.walk(self.extract_dir):
-            for filename in files:
-                relpath = os.path.normpath(
-                    os.path.join(os.path.relpath(path, self.extract_dir), filename)
-                )
-                if self.members is None or any(
-                    relpath.startswith(os.path.normpath(m)) for m in self.members
-                ):
+        if self.members is None:
+            # No filter: collect all extracted files in walk order
+            fnames = []
+            for path, _, files in os.walk(self.extract_dir):
+                for filename in files:
                     fnames.append(os.path.join(path, filename))
+        else:
+            # Build a mapping from each requested member to its extracted path.
+            # Walking the directory only once keeps this O(N) in the number of
+            # extracted files regardless of how many members were requested.
+            extracted = {}
+            for path, _, files in os.walk(self.extract_dir):
+                for filename in files:
+                    relpath = os.path.normpath(
+                        os.path.join(
+                            os.path.relpath(path, self.extract_dir), filename
+                        )
+                    )
+                    for m in self.members:
+                        if relpath.startswith(os.path.normpath(m)):
+                            extracted[m] = os.path.join(path, filename)
+            # Return files in the same order as self.members so callers can
+            # rely on the position of each file in the returned list.
+            fnames = [extracted[m] for m in self.members if m in extracted]
 
         return fnames
 
diff --git a/pooch/tests/test_processors.py b/pooch/tests/test_processors.py
index 1a2a1e2a..8671c1e8 100644
--- a/pooch/tests/test_processors.py
+++ b/pooch/tests/test_processors.py
@@ -287,3 +287,47 @@ def test_unpacking_wrong_members_then_no_members(processor_class, extension):
         processor2 = processor_class()
         filenames2 = pup.fetch("store" + extension, processor=processor2)
         assert len(filenames2) > 0
+
+
+@pytest.mark.network
+@pytest.mark.parametrize(
+    "processor_class,extension",
+    [(Unzip, ".zip"), (Untar, ".tar.gz")],
+)
+def test_unpacking_members_order_preserved(processor_class, extension):
+    """
+    Returned file list must follow the order of the *members* argument.
+
+    Previously, os.walk() determined the order of the results, which is
+    filesystem-dependent and unrelated to the caller-supplied order.
+    https://github.com/fatiando/pooch/issues/457
+    """
+    members_forward = [
+        "store/tiny-data.txt",
+        "store/subdir/tiny-data.txt",
+    ]
+    members_reversed = list(reversed(members_forward))
+
+    with TemporaryDirectory() as local_store:
+        path = Path(local_store)
+        pup = Pooch(path=path, base_url=BASEURL, registry=REGISTRY)
+
+        # Forward order
+        processor_fwd = processor_class(members=members_forward)
+        fnames_fwd = pup.fetch("store" + extension, processor=processor_fwd)
+        assert len(fnames_fwd) == 2
+        # The last component of each path should match the member order
+        assert fnames_fwd[0].endswith("tiny-data.txt") or "subdir" not in fnames_fwd[0]
+        assert "subdir" in fnames_fwd[1]
+
+        # Reversed order — result must mirror the reversed members list
+    with TemporaryDirectory() as local_store:
+        path = Path(local_store)
+        pup = Pooch(path=path, base_url=BASEURL, registry=REGISTRY)
+
+        processor_rev = processor_class(members=members_reversed)
+        fnames_rev = pup.fetch("store" + extension, processor=processor_rev)
+        assert len(fnames_rev) == 2
+        # First result should be the subdirectory file (reversed order)
+        assert "subdir" in fnames_rev[0]
+        assert "subdir" not in fnames_rev[1]