diff --git a/src/dvc_data/index/collect.py b/src/dvc_data/index/collect.py
index 14e44ef4..6493b16a 100644
--- a/src/dvc_data/index/collect.py
+++ b/src/dvc_data/index/collect.py
@@ -94,7 +94,7 @@ def collect(  # noqa: C901, PLR0912
             cache = storage_info.cache if storage != "cache" else None
             remote = storage_info.remote if storage != "remote" else None
 
-            if not data:
+            if not data or (push and data.read_only):
                 continue
 
             try:
diff --git a/src/dvc_data/index/index.py b/src/dvc_data/index/index.py
index f89d93b8..0a591012 100644
--- a/src/dvc_data/index/index.py
+++ b/src/dvc_data/index/index.py
@@ -144,8 +144,9 @@ def close(self):
 
 
 class Storage(ABC):
-    def __init__(self, key: "DataIndexKey"):
+    def __init__(self, key: "DataIndexKey", read_only: bool = False):
         self.key = key
+        self.read_only = read_only
 
     @property
     @abstractmethod
@@ -176,10 +177,11 @@ def __init__(
         key: "DataIndexKey",
         odb: "HashFileDB",
         index: Optional["DataIndex"] = None,
+        read_only: bool = False,
     ):
         self.odb = odb
         self.index = index
-        super().__init__(key)
+        super().__init__(key, read_only=read_only)
 
     @property
     def fs(self):
@@ -235,12 +237,13 @@ def __init__(
         path: "str",
         index: Optional["DataIndex"] = None,
         prefix: Optional["DataIndexKey"] = None,
+        read_only: bool = False,
     ):
         self._fs = fs
         self._path = path
         self.index = index
         self.prefix = prefix if prefix is not None else key
-        super().__init__(key)
+        super().__init__(key, read_only=read_only)
 
     @property
     def fs(self):