diff --git a/swh/dataset/journalprocessor.py b/swh/dataset/journalprocessor.py
--- a/swh/dataset/journalprocessor.py
+++ b/swh/dataset/journalprocessor.py
@@ -71,7 +71,8 @@
         self.count = 0
         try:
             self.handle_committed_offsets()
-            super().process(*args, **kwargs)
+            if self.assignment:
+                super().process(*args, **kwargs)
         except EOFError:
             pass
         finally:
@@ -104,7 +105,7 @@
                 self.assignment = [
                     pid for pid in self.assignment if pid != partition_id
                 ]
-                self.subscribe()
+                self.subscribe()  # Actually, unsubscripted from the partition_id
 
         if not self.assignment:
             raise EOFError
@@ -175,7 +176,8 @@
             def fetch_insert_partition_id(partition_id):
                 tp = TopicPartition(topic_name, partition_id)
                 (lo, hi) = client.consumer.get_watermark_offsets(tp)
-                self.offsets[partition_id] = (lo, hi)
+                if lo != hi:
+                    self.offsets[partition_id] = (lo, hi)
 
             with concurrent.futures.ThreadPoolExecutor(
                 max_workers=self.processes