diff --git a/dvc/output/base.py b/dvc/output/base.py
index 420fc65166..d97887e5c3 100644
--- a/dvc/output/base.py
+++ b/dvc/output/base.py
@@ -1,15 +1,13 @@
 import logging
 import os
 from copy import copy
-from typing import Dict, Type
+from typing import Type
 from urllib.parse import urlparse
 
-from funcy import cached_property, project
 from voluptuous import Any
 
 import dvc.prompt as prompt
 from dvc.cache import NamedCache
-from dvc.env import DVCLIVE_PATH, DVCLIVE_REPORT, DVCLIVE_SUMMARY
 from dvc.exceptions import (
     CheckoutError,
     CollectCacheError,
@@ -607,22 +605,3 @@ def merge(self, ancestor, other):
         self.hash_info = self.cache.merge(
             ancestor_info, self.hash_info, other.hash_info
         )
-
-    @cached_property
-    def env(self) -> Dict[str, str]:
-        if self.live:
-            from dvc.schema import LIVE_PROPS
-
-            env = {DVCLIVE_PATH: str(self.path_info)}
-            if isinstance(self.live, dict):
-
-                config = project(self.live, LIVE_PROPS)
-
-                env[DVCLIVE_SUMMARY] = str(
-                    int(config.get(BaseOutput.PARAM_LIVE_SUMMARY, True))
-                )
-                env[DVCLIVE_REPORT] = str(
-                    int(config.get(BaseOutput.PARAM_LIVE_REPORT, True))
-                )
-            return env
-        return {}
diff --git a/dvc/stage/__init__.py b/dvc/stage/__init__.py
index e6ca0cf56d..990d3016dc 100644
--- a/dvc/stage/__init__.py
+++ b/dvc/stage/__init__.py
@@ -255,13 +255,41 @@ def is_checkpoint(self):
         """
         return any(out.checkpoint for out in self.outs)
 
-    @property
-    def env(self) -> Env:
+    def _read_env(self, out, checkpoint_func=None) -> Env:
         env: Env = {}
+        if out.live:
+            from dvc.env import DVCLIVE_PATH, DVCLIVE_REPORT, DVCLIVE_SUMMARY
+            from dvc.output import BaseOutput
+            from dvc.schema import LIVE_PROPS
+
+            env[DVCLIVE_PATH] = str(out.path_info)
+            if isinstance(out.live, dict):
+                config = project(out.live, LIVE_PROPS)
+
+                env[DVCLIVE_SUMMARY] = str(
+                    int(config.get(BaseOutput.PARAM_LIVE_SUMMARY, True))
+                )
+                env[DVCLIVE_REPORT] = str(
+                    int(config.get(BaseOutput.PARAM_LIVE_REPORT, True))
+                )
+        elif out.checkpoint and checkpoint_func:
+            from dvc.env import DVC_CHECKPOINT
+
+            env.update({DVC_CHECKPOINT: "1"})
+        return env
+
+    def env(self, checkpoint_func=None) -> Env:
+        from dvc.env import DVC_ROOT
+
+        env: Env = {}
+        if self.repo:
+            env.update({DVC_ROOT: self.repo.root_dir})
+
         for out in self.outs:
-            if any(out.env.keys() and env.keys()):
+            current = self._read_env(out, checkpoint_func=checkpoint_func)
+            if set(env.keys()).intersection(set(current.keys())):
                 raise DvcException("Duplicated env variable")
-            env.update(out.env)
+            env.update(current)
         return env
 
     def changed_deps(self):
diff --git a/dvc/stage/run.py b/dvc/stage/run.py
index d747409e17..afdb6f31bf 100644
--- a/dvc/stage/run.py
+++ b/dvc/stage/run.py
@@ -5,7 +5,6 @@
 import threading
 from contextlib import contextmanager
 
-from dvc.env import DVC_CHECKPOINT, DVC_ROOT
 from dvc.utils import fix_env
 
 from .decorators import relock_repo, unlocked_repo
@@ -52,11 +51,8 @@ def _enforce_cmd_list(cmd):
 
 def prepare_kwargs(stage, checkpoint_func=None):
     kwargs = {"cwd": stage.wdir, "env": fix_env(None), "close_fds": True}
-    if checkpoint_func:
-        # indicate that checkpoint cmd is being run inside DVC
-        kwargs["env"].update(_checkpoint_env(stage))
 
-    kwargs["env"].update(stage.env)
+    kwargs["env"].update(stage.env(checkpoint_func=checkpoint_func))
 
     # NOTE: when you specify `shell=True`, `Popen` [1] will default to
     # `/bin/sh` on *nix and will add ["/bin/sh", "-c"] to your command.
@@ -143,10 +139,6 @@ def run_stage(stage, dry=False, force=False, checkpoint_func=None, **kwargs):
     run(stage, dry=dry, checkpoint_func=checkpoint_func)
 
 
-def _checkpoint_env(stage):
-    return {DVC_CHECKPOINT: "1", DVC_ROOT: stage.repo.root_dir}
-
-
 @contextmanager
 def checkpoint_monitor(stage, callback_func, proc, killed):
     if not callback_func: