diff --git a/dvc/repo/data.py b/dvc/repo/data.py
index 629f64c0b..17b318d52 100644
--- a/dvc/repo/data.py
+++ b/dvc/repo/data.py
@@ -94,7 +94,7 @@ def _diff(
             with_missing
             and change.old
             and change.old.hash_info
-            and not old.storage_map[change.key].cache.exists(change.old.hash_info.value)
+            and not old.storage_map.cache_exists(change.old)
         ):
             # NOTE: emulating previous behaviour
             _add_change("not_in_cache", change)
diff --git a/dvc/repo/imports.py b/dvc/repo/imports.py
index c454772a8..3405cfa09 100644
--- a/dvc/repo/imports.py
+++ b/dvc/repo/imports.py
@@ -115,7 +115,7 @@ def save_imports(
                 desc="Downloading imports from source",
                 unit="files",
             ) as cb:
-                checkout(data_view, tmpdir, cache.fs, callback=cb)
+                checkout(data_view, tmpdir, cache.fs, callback=cb, storage="data")
             md5(data_view)
             save(data_view, odb=cache, hardlink=True)
 
diff --git a/dvc/repo/index.py b/dvc/repo/index.py
index a97c99367..e6996b22f 100644
--- a/dvc/repo/index.py
+++ b/dvc/repo/index.py
@@ -286,7 +286,7 @@ class Index:
         from collections import defaultdict
 
         from dvc.config import NoRemoteError
-        from dvc_data.index import DataIndex, DataIndexEntry, Storage
+        from dvc_data.index import DataIndex, DataIndexEntry, FileStorage, ObjectStorage
 
         by_workspace: dict = defaultdict(DataIndex)
 
@@ -306,9 +306,22 @@ class Index:
                 hash_info=out.hash_info,
             )
 
-            storage = Storage(odb=out.odb, cache=out.cache)
+            storage_map = data_index.storage_map
+            if out.odb:
+                storage_map.add_data(ObjectStorage(key, out.odb))
+            storage_map.add_cache(ObjectStorage(key, out.cache))
             try:
-                storage.remote = self.repo.cloud.get_remote_odb(out.remote)
+                remote = self.repo.cloud.get_remote(out.remote)
+                if remote.fs.version_aware:
+                    storage_map.add_remote(
+                        FileStorage(
+                            key=key,
+                            fs=remote.fs,
+                            path=remote.fs.path.join(remote.path, *key),
+                        )
+                    )
+                else:
+                    storage_map.add_remote(ObjectStorage(key, remote.odb))
             except NoRemoteError:
                 pass
 
@@ -318,12 +331,9 @@ class Index:
                 entry.meta = dep.meta
                 entry.hash_info = dep.hash_info
 
-                storage.odb = dep.odb
-                storage.fs = dep.fs
-                storage.path = dep.fs_path
+                storage_map.add_data(FileStorage(key, dep.fs, dep.fs_path))
 
             data_index.add(entry)
-            data_index.storage_map[key] = storage
 
         return dict(by_workspace)
 
diff --git a/dvc/repo/worktree.py b/dvc/repo/worktree.py
index a60baee4b..4dd13e4ad 100644
--- a/dvc/repo/worktree.py
+++ b/dvc/repo/worktree.py
@@ -119,12 +119,6 @@ def fetch_worktree(
     ):
         remote_obj = _get_remote(repo, remote_name, remote, "fetch")
         index = view.data["repo"]
-        for key in index.storage_map:
-            storage = index.storage_map[key]
-            storage.fs = remote_obj.fs
-            storage.path = remote_obj.fs.path.join(remote_obj.path, *key)
-            index.storage_map[key] = storage
-
         total = len(index)
         with Callback.as_tqdm_callback(
             unit="file",
@@ -132,7 +126,7 @@ def fetch_worktree(
             disable=total == 0,
         ) as cb:
             cb.set_size(total)
-            transferred += save(index, callback=cb, jobs=jobs)
+            transferred += save(index, callback=cb, jobs=jobs, storage="remote")
     return transferred
 
 
@@ -357,6 +351,7 @@ def _fetch_out_changes(
             update_meta=False,
             meta_only=True,
             meta_cmp_key=partial(_meta_checksum, remote.fs),
+            storage="data",
             callback=cb,
         )
         out.save()
diff --git a/pyproject.toml b/pyproject.toml
index a04015b2b..400437a7d 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -53,7 +53,7 @@ dependencies = [
     "dvc-render>=0.1.2",
     "dvc-task==0.1.11",
     "dvclive>=1.2.2",
-    "dvc-data==0.39.3",
+    "dvc-data==0.40.1",
     "dvc-http",
     "hydra-core>=1.1.0",
     "iterative-telemetry==0.0.7",