diff --git a/pooch/processors.py b/pooch/processors.py
index a5f7f20..dbe1a91 100644
--- a/pooch/processors.py
+++ b/pooch/processors.py
@@ -79,17 +79,36 @@ class ExtractorProcessor:  # pylint: disable=too-few-public-methods
         else:
             archive_dir = fname.rsplit(os.path.sep, maxsplit=1)[0]
             self.extract_dir = os.path.join(archive_dir, self.extract_dir)
-        if action in ("update", "download") or not os.path.exists(self.extract_dir):
+        if (
+            (action in ("update", "download"))
+            or (not os.path.exists(self.extract_dir))
+            or (
+                (self.members is not None)
+                and (
+                    not all(
+                        os.path.exists(os.path.join(self.extract_dir, m))
+                        for m in self.members
+                    )
+                )
+            )
+        ):
             # Make sure that the folder with the extracted files exists
             os.makedirs(self.extract_dir, exist_ok=True)
             self._extract_file(fname, self.extract_dir)
+
         # Get a list of all file names (including subdirectories) in our folder
-        # of unzipped files.
-        fnames = [
-            os.path.join(path, fname)
-            for path, _, files in os.walk(self.extract_dir)
-            for fname in files
-        ]
+        # 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
+                ):
+                    fnames.append(os.path.join(path, filename))
+
         return fnames
 
     def _extract_file(self, fname, extract_dir):
@@ -153,7 +172,9 @@ class Unzip(ExtractorProcessor):  # pylint: disable=too-few-public-methods
                     # Based on:
                     # https://stackoverflow.com/questions/8008829/extract-only-a-single-directory-from-tar
                     subdir_members = [
-                        name for name in zip_file.namelist() if name.startswith(member)
+                        name
+                        for name in zip_file.namelist()
+                        if os.path.normpath(name).startswith(os.path.normpath(member))
                     ]
                     # Extract the data file from within the archive
                     zip_file.extractall(members=subdir_members, path=extract_dir)
@@ -216,7 +237,9 @@ class Untar(ExtractorProcessor):  # pylint: disable=too-few-public-methods
                     subdir_members = [
                         info
                         for info in tar_file.getmembers()
-                        if info.name.startswith(member)
+                        if os.path.normpath(info.name).startswith(
+                            os.path.normpath(member)
+                        )
                     ]
                     # Extract the data file from within the archive
                     tar_file.extractall(members=subdir_members, path=extract_dir)

diff --git a/pooch/tests/test_processors.py b/pooch/tests/test_processors.py
index 82462c3..578b478 100644
--- a/pooch/tests/test_processors.py
+++ b/pooch/tests/test_processors.py
@@ -169,6 +169,57 @@ def test_unpacking(processor_class, extension, target_path, archive, members):
             check_tiny_data(fname)
 
 
+@pytest.mark.network
+@pytest.mark.parametrize(
+    "processor_class,extension",
+    [(Unzip, ".zip"), (Untar, ".tar.gz")],
+)
+def test_multiple_unpacking(processor_class, extension):
+    "Test that multiple subsequent calls to a processor yield correct results"
+
+    with TemporaryDirectory() as local_store:
+        pup = Pooch(path=Path(local_store), base_url=BASEURL, registry=REGISTRY)
+
+        # Do a first fetch with the one member only
+        processor1 = processor_class(members=["store/tiny-data.txt"])
+        filenames1 = pup.fetch("store" + extension, processor=processor1)
+        assert len(filenames1) == 1
+        check_tiny_data(filenames1[0])
+
+        # Do a second fetch with the other member
+        processor2 = processor_class(
+            members=["store/tiny-data.txt", "store/subdir/tiny-data.txt"]
+        )
+        filenames2 = pup.fetch("store" + extension, processor=processor2)
+        assert len(filenames2) == 2
+        check_tiny_data(filenames2[0])
+        check_tiny_data(filenames2[1])
+
+        # Do a third fetch, again with one member and assert
+        # that only this member was returned
+        filenames3 = pup.fetch("store" + extension, processor=processor1)
+        assert len(filenames3) == 1
+        check_tiny_data(filenames3[0])
+
+
+@pytest.mark.network
+@pytest.mark.parametrize(
+    "processor_class,extension",
+    [(Unzip, ".zip"), (Untar, ".tar.gz")],
+)
+def test_unpack_members_with_leading_dot(processor_class, extension):
+    "Test that unpack members can also be specifed both with a leading ./"
+
+    with TemporaryDirectory() as local_store:
+        pup = Pooch(path=Path(local_store), base_url=BASEURL, registry=REGISTRY)
+
+        # Do a first fetch with the one member only
+        processor1 = processor_class(members=["./store/tiny-data.txt"])
+        filenames1 = pup.fetch("store" + extension, processor=processor1)
+        assert len(filenames1) == 1
+        check_tiny_data(filenames1[0])
+
+
 def _check_logs(log_file, expected_lines):
     """
     Assert that the lines in the log match the expected ones.

Repeated fetching with differing Untar processors yields wrong results
**Description of the problem:**

The use case is to download an archive, unpack it and get access to individual files within that archive through the `members` parameter of `pooch.Untar`. When doing so, the `members` argument of the first call is cached for subsequent calls, which prohibits a usage pattern that looks otherwise valid. `pooch.Unzip` is likely to have the same issue.

**Full code that generated the error**

```python
import pooch
P = pooch.create(
    path=pooch.os_cache("test"),
    base_url="https://github.com/ssciwr/afwizard-test-data/releases/download/2022-06-09/",
    registry={
        "data.tar.gz": "sha256:fae90a3cf758e2346b81fa0e3b005f2914d059ca182202a1de8f627b1ec0c160"
    }
)
files1 = P.fetch("data.tar.gz", processor=pooch.Untar(members=["testfilter1.json"]))
files2 = P.fetch("data.tar.gz", processor=pooch.Untar(members=["testfilter2.json"]))
assert files1[0] != files2[0]
```

**Full error message**

Above assertion fails.
pooch/tests/test_processors.py::test_multiple_unpacking[Unzip-.zip] pooch/tests/test_processors.py::test_multiple_unpacking[Untar-.tar.gz] pooch/tests/test_processors.py::test_unpack_members_with_leading_dot[Unzip-.zip] pooch/tests/test_processors.py::test_unpack_members_with_leading_dot[Untar-.tar.gz]
pooch/tests/test_processors.py::test_decompress[auto] pooch/tests/test_processors.py::test_decompress[lzma] pooch/tests/test_processors.py::test_decompress[xz] pooch/tests/test_processors.py::test_decompress[bz2] pooch/tests/test_processors.py::test_decompress[gz] pooch/tests/test_processors.py::test_decompress[name] pooch/tests/test_processors.py::test_decompress_fails pooch/tests/test_processors.py::test_extractprocessor_fails pooch/tests/test_processors.py::test_unpacking[Unzip-single_file-default_path] pooch/tests/test_processors.py::test_unpacking[Unzip-single_file-custom_path] pooch/tests/test_processors.py::test_unpacking[Unzip-archive_all-default_path] pooch/tests/test_processors.py::test_unpacking[Unzip-archive_all-custom_path] pooch/tests/test_processors.py::test_unpacking[Unzip-archive_file-default_path] pooch/tests/test_processors.py::test_unpacking[Unzip-archive_file-custom_path] pooch/tests/test_processors.py::test_unpacking[Unzip-archive_subdir_file-default_path] pooch/tests/test_processors.py::test_unpacking[Unzip-archive_subdir_file-custom_path] pooch/tests/test_processors.py::test_unpacking[Unzip-archive_subdir-default_path] pooch/tests/test_processors.py::test_unpacking[Unzip-archive_subdir-custom_path] pooch/tests/test_processors.py::test_unpacking[Unzip-archive_multiple-default_path] pooch/tests/test_processors.py::test_unpacking[Unzip-archive_multiple-custom_path] pooch/tests/test_processors.py::test_unpacking[Untar-single_file-default_path] pooch/tests/test_processors.py::test_unpacking[Untar-single_file-custom_path] pooch/tests/test_processors.py::test_unpacking[Untar-archive_all-default_path] pooch/tests/test_processors.py::test_unpacking[Untar-archive_all-custom_path] pooch/tests/test_processors.py::test_unpacking[Untar-archive_file-default_path] pooch/tests/test_processors.py::test_unpacking[Untar-archive_file-custom_path] pooch/tests/test_processors.py::test_unpacking[Untar-archive_subdir_file-default_path] pooch/tests/test_processors.py::test_unpacking[Untar-archive_subdir_file-custom_path] pooch/tests/test_processors.py::test_unpacking[Untar-archive_subdir-default_path] pooch/tests/test_processors.py::test_unpacking[Untar-archive_subdir-custom_path] pooch/tests/test_processors.py::test_unpacking[Untar-archive_multiple-default_path] pooch/tests/test_processors.py::test_unpacking[Untar-archive_multiple-custom_path]