diff --git a/coverage/control.py b/coverage/control.py
index 0d50783cd..7f94f8150 100644
--- a/coverage/control.py
+++ b/coverage/control.py
@@ -559,12 +559,17 @@ def load(self) -> None:
 
     def _init_for_start(self) -> None:
         """Initialization for start()"""
+        self.orig_dir = None
         # Construct the collector.
         concurrency: list[str] = self.config.concurrency or []
+        coverpath_file = os.path.join(os.getcwd(), ".coverpath")
         if "multiprocessing" in concurrency:
             if self.config.config_file is None:
                 raise ConfigError("multiprocessing requires a configuration file")
             patch_multiprocessing(rcfile=self.config.config_file)
+            if os.path.exists(coverpath_file):
+                with open(coverpath_file, encoding="utf-8") as cf:
+                    self.orig_dir = cf.read().strip()
 
         dycon = self.config.dynamic_context
         if not dycon or dycon == "none":
@@ -656,6 +661,8 @@ def _init_for_start(self) -> None:
 
     def _init_data(self, suffix: str | bool | None) -> None:
         """Create a data file if we don't have one yet."""
+        if not hasattr(self, "orig_dir"):
+            self.orig_dir = None
         if self._data is None:
             # Create the data file.  We do this at construction time so that the
             # data file will be written into the directory where the process
@@ -667,6 +674,7 @@ def _init_data(self, suffix: str | bool | None) -> None:
                 warn=self._warn,
                 debug=self._debug,
                 no_disk=self._no_disk,
+                orig_dir=self.orig_dir,
             )
             self._data_to_close.append(self._data)
 
diff --git a/coverage/multiproc.py b/coverage/multiproc.py
index 1c2d2f7d7..fa1ae524d 100644
--- a/coverage/multiproc.py
+++ b/coverage/multiproc.py
@@ -113,6 +113,8 @@ def get_preparation_data_with_stowaway(name: str) -> dict[str, Any]:
             """Get the original preparation data, and also insert our stowaway."""
             d = original_get_preparation_data(name)
             d["stowaway"] = Stowaway(rcfile)
+            with open(".coverpath", "w", encoding="utf-8") as cpath:
+                cpath.write(d["orig_dir"])
             return d
 
         spawn.get_preparation_data = get_preparation_data_with_stowaway
diff --git a/coverage/sqldata.py b/coverage/sqldata.py
index 3c958f405..cc7421277 100644
--- a/coverage/sqldata.py
+++ b/coverage/sqldata.py
@@ -11,6 +11,7 @@
 import glob
 import itertools
 import os
+import shutil
 import random
 import socket
 import sqlite3
@@ -232,6 +233,8 @@ def __init__(
         no_disk: bool = False,
         warn: TWarnFn | None = None,
         debug: TDebugCtl | None = None,
+        *,
+        orig_dir = None,
     ) -> None:
         """Create a :class:`CoverageData` object to hold coverage-measured data.
 
@@ -271,6 +274,7 @@ def __init__(
         self._current_context: str | None = None
         self._current_context_id: int | None = None
         self._query_context_ids: list[int] | None = None
+        self.orig_dir = orig_dir
 
     __repr__ = auto_repr
 
@@ -898,6 +902,11 @@ def read(self) -> None:
     def write(self) -> None:
         """Ensure the data is written to the data file."""
         self._debug_dataio("Writing (no-op) data file", self._filename)
+        if self.orig_dir is not None:
+            for fname in os.listdir():
+                if fname.startswith(".coverage."):
+                    if not os.path.exists(os.path.join(self.orig_dir, fname)):
+                        shutil.copy(fname, self.orig_dir)
 
     def _start_using(self) -> None:
         """Call this before using the database at all."""
diff --git a/tests/test_concurrency.py b/tests/test_concurrency.py
index dd761e377..858bc6bc9 100644
--- a/tests/test_concurrency.py
+++ b/tests/test_concurrency.py
@@ -13,6 +13,7 @@
 import re
 import sys
 import threading
+import concurrent.futures
 import time
 
 from types import ModuleType
@@ -412,6 +413,32 @@ def work(x):
         return sum_range((x+1)*100)
     """
 
+MULTI_CODE_DIR_CHANGE = """
+    import os
+    import tempfile
+    from concurrent.futures import ProcessPoolExecutor
+
+    def add(a, b):
+        return a + b
+
+    def probe_dispatcher():
+        orig_dir = os.getcwd()
+        with tempfile.TemporaryDirectory() as temp_dir:
+            os.chdir(temp_dir)
+            dispatcher()
+        os.chdir(orig_dir)
+
+    def dispatcher():
+        futures = []
+        with ProcessPoolExecutor({NPROCS}) as executor:
+            futures.append(executor.submit(add, 2, 2))
+        for future in futures:
+            future.result()
+
+    if __name__ == "__main__":
+        probe_dispatcher()
+    """
+
 MULTI_CODE = """
     # Above this will be a definition of work().
     import multiprocessing
@@ -522,6 +549,20 @@ def test_multiprocessing_simple(self, start_method: str) -> None:
             start_method=start_method,
         )
 
+    def test_gh_2065(self, start_method: str) -> None:
+        nprocs = 1
+        upto = 30
+        code = (MULTI_CODE_DIR_CHANGE).format(NPROCS=nprocs, UPTO=upto)
+        total = 0
+        expected_out = f""
+        self.try_multiprocessing_code(
+            code,
+            expected_out,
+            concurrent.futures,
+            nprocs,
+            start_method=start_method,
+        )
+
     def test_multiprocessing_append(self, start_method: str) -> None:
         nprocs = 3
         upto = 30