diff --git a/swh/dataset/luigi.py b/swh/dataset/luigi.py
--- a/swh/dataset/luigi.py
+++ b/swh/dataset/luigi.py
@@ -39,6 +39,7 @@
 ``stamps`` files are written after corresponding directories are written.
 Their presence indicates the corresponding directory was fully generated/copied.
 This allows skipping work that was already done, while ignoring interrupted jobs.
+They are omitted after the initial export (ie. when downloading to/from other machines).
 
 ``meta/export.json`` contains information about the dataset, for provenance tracking.
 For example:
@@ -111,7 +112,7 @@
 # control
 import enum
 from pathlib import Path
-from typing import Hashable, Iterator, List, TypeVar
+from typing import Hashable, Iterator, List, TypeVar, Union
 
 import luigi
 
@@ -212,6 +213,19 @@
     ]
 
 
+def _export_metadata_has_object_types(
+    export_metadata: Union[luigi.LocalTarget, "luigi.contrib.s3.S3Target"],
+    object_types: List[ObjectType],
+) -> bool:
+    import json
+
+    with export_metadata.open() as fd:
+        meta = json.load(fd)
+    return set(meta["object_type"]) >= {
+        object_type.name for object_type in object_types
+    }
+
+
 class ExportGraph(luigi.Task):
     """Exports the entire graph to the local filesystem.
 
@@ -347,25 +361,20 @@
 
     def output(self) -> List[luigi.Target]:
         """Returns stamp and meta paths on S3."""
-        return self._stamps() + [self._meta()]
-
-    def _stamps(self):
-        import luigi.contrib.s3
-
-        return [
-            luigi.contrib.s3.S3Target(f"{self.s3_export_path}/{path}")
-            for path in stamps_paths(self.formats, self.object_types)
-        ]
+        return [self._meta()]
 
     def _meta(self):
         import luigi.contrib.s3
 
         return luigi.contrib.s3.S3Target(f"{self.s3_export_path}/meta/export.json")
 
+    def complete(self) -> bool:
+        return super().complete() and _export_metadata_has_object_types(
+            self._meta(), self.object_types
+        )
+
     def run(self) -> None:
-        """Copies all files: first the export itself, then stamps, then
-        :file:`meta.json`.
-        """
+        """Copies all files: first the export itself, then :file:`meta.json`."""
         import os
 
         import luigi.contrib.s3
@@ -390,13 +399,6 @@
                         local_dir / file_, f"{s3_dir}/{file_}", ACL="public-read"
                     )
 
-            for stamp in stamps_paths(self.formats, self.object_types):
-                client.put_multipart(
-                    self.local_export_path / stamp,
-                    f"{self.s3_export_path}/{stamp}",
-                    ACL="public-read",
-                )
-
         client.put(
             self.local_export_path / "meta" / "export.json",
             self._meta().path,
@@ -438,28 +440,25 @@
         ]
 
     def output(self) -> List[luigi.Target]:
-        """Returns stamp and meta paths on S3."""
-        return self._stamps() + [self._meta()]
+        """Returns stamp and meta paths on the local filesystem."""
+        return [self._meta()]
 
-    def _stamps(self):
-        return [
-            luigi.LocalTarget(self.local_export_path / path)
-            for path in stamps_paths(self.formats, self.object_types)
-        ]
+    def complete(self) -> bool:
+        return super().complete() and _export_metadata_has_object_types(
+            self._meta(), self.object_types
+        )
 
     def _meta(self):
         return luigi.LocalTarget(self.local_export_path / "meta" / "export.json")
 
     def run(self) -> None:
-        """Copies all files: first the export itself, then stamps, then
-        :file:`meta.json`.
-        """
+        """Copies all files: first the export itself, then :file:`meta.json`."""
         import luigi.contrib.s3
         import tqdm
 
         client = luigi.contrib.s3.S3Client()
 
-        # recursively copy local files to S3, and end with stamps and export metadata
+        # recursively copy local files to S3, and end with export metadata
         for format_ in self.formats:
             for object_type in self.object_types:
                 local_dir = self.local_export_path / format_.name / object_type.name
@@ -479,14 +478,6 @@
                         str(local_dir / file_),
                     )
 
-            for stamp in stamps_paths(self.formats, self.object_types):
-                stamp_path = self.local_export_path / stamp
-                stamp_path.parent.mkdir(parents=True, exist_ok=True)
-                client.get(
-                    f"{self.s3_export_path}/{stamp}",
-                    str(stamp_path),
-                )
-
         export_json_path = self.local_export_path / "meta" / "export.json"
         export_json_path.parent.mkdir(exist_ok=True)
         client.get(