diff --git a/asyncssh/process.py b/asyncssh/process.py
index b4bea25..33a0cdb 100644
--- a/asyncssh/process.py
+++ b/asyncssh/process.py
@@ -30,8 +30,8 @@
 import socket
 import stat
 from types import TracebackType
-from typing import Any, AnyStr, Callable, Dict, Generic, IO
-from typing import Iterable, Mapping, Optional, Set, TextIO
+from typing import Any, AnyStr, Awaitable, Callable, Dict, Generic, IO
+from typing import Iterable, List, Mapping, Optional, Set, TextIO
 from typing import Tuple, Type, TypeVar, Union, cast
 from typing_extensions import Protocol
 
@@ -308,14 +308,33 @@ def __init__(self, process: 'SSHProcess[AnyStr]',
                  encoding: Optional[str], errors: str):
         super().__init__(encoding, errors, hasattr(file, 'encoding'))
 
-        self._conn = process.channel.get_connection()
+        self._process: 'SSHProcess[AnyStr]' = process
         self._file = file
         self._needs_close = needs_close
+        self._queue: asyncio.Queue[Optional[AnyStr]] = asyncio.Queue()
+        self._write_task: Optional[asyncio.Task[None]] = \
+            process.channel.get_connection().create_task(self._writer())
+
+    async def _writer(self) -> None:
+        """Process writes to the file"""
+
+        while True:
+            data = await self._queue.get()
+
+            if data is None:
+                self._queue.task_done()
+                break
+
+            await self._file.write(self.encode(data))
+            self._queue.task_done()
+
+        if self._needs_close:
+            await self._file.close()
 
     def write(self, data: AnyStr) -> None:
         """Write data to the file"""
 
-        self._conn.create_task(self._file.write(self.encode(data)))
+        self._queue.put_nowait(data)
 
     def write_eof(self) -> None:
         """Close output file when end of file is received"""
@@ -325,8 +344,10 @@ def write_eof(self) -> None:
     def close(self) -> None:
         """Stop forwarding data to the file"""
 
-        if self._needs_close:
-            self._conn.create_task(self._file.close())
+        if self._write_task:
+            self._write_task = None
+            self._queue.put_nowait(None)
+            self._process.add_cleanup_task(self._queue.join())
 
 
 class _PipeReader(_UnicodeReader[AnyStr], asyncio.BaseProtocol):
@@ -721,6 +742,8 @@ class SSHProcess(SSHStreamSession, Generic[AnyStr]):
     def __init__(self, *args) -> None:
         super().__init__(*args)
 
+        self._cleanup_tasks: List[Awaitable[None]] = []
+
         self._readers: Dict[Optional[int], _ReaderProtocol] = {}
         self._send_eof: Dict[Optional[int], bool] = {}
 
@@ -729,6 +752,20 @@ def __init__(self, *args) -> None:
 
         self._paused_write_streams: Set[Optional[int]] = set()
 
+    async def __aenter__(self) -> 'SSHProcess[AnyStr]':
+        """Allow SSHProcess to be used as an async context manager"""
+
+        return self
+
+    async def __aexit__(self, _exc_type: Optional[Type[BaseException]],
+                        _exc_value: Optional[BaseException],
+                        _traceback: Optional[TracebackType]) -> bool:
+        """Wait for a full channel close when exiting the async context"""
+
+        self.close()
+        await self.wait_closed()
+        return False
+
     @property
     def channel(self) -> SSHChannel[AnyStr]:
         """The channel associated with the process"""
@@ -931,6 +968,11 @@ def _should_pause_reading(self) -> bool:
         return bool(self._paused_write_streams) or \
             super()._should_pause_reading()
 
+    def add_cleanup_task(self, task: Awaitable[None]) -> None:
+        """Add a task to run when the process exits"""
+
+        self._cleanup_tasks.append(task)
+
     def connection_lost(self, exc: Optional[Exception]) -> None:
         """Handle a close of the SSH channel"""
 
@@ -1091,6 +1133,9 @@ async def wait_closed(self) -> None:
         assert self._chan is not None
         await self._chan.wait_closed()
 
+        for task in self._cleanup_tasks:
+            await task
+
 
 class SSHClientProcess(SSHProcess[AnyStr], SSHClientStreamSession[AnyStr]):
     """SSH client process handler"""
@@ -1105,20 +1150,6 @@ def __init__(self) -> None:
         self._stdout: Optional[SSHReader[AnyStr]] = None
         self._stderr: Optional[SSHReader[AnyStr]] = None
 
-    async def __aenter__(self) -> 'SSHClientProcess[AnyStr]':
-        """Allow SSHProcess to be used as an async context manager"""
-
-        return self
-
-    async def __aexit__(self, _exc_type: Optional[Type[BaseException]],
-                        _exc_value: Optional[BaseException],
-                        _traceback: Optional[TracebackType]) -> bool:
-        """Wait for a full channel close when exiting the async context"""
-
-        self.close()
-        await self._chan.wait_closed()
-        return False
-
     def _collect_output(self, datatype: DataType = None) -> AnyStr:
         """Return output from the process"""
 
@@ -1333,7 +1364,7 @@ async def communicate(self, input: Optional[AnyStr] = None) -> \
             self._chan.write(input)
             self._chan.write_eof()
 
-        await self._chan.wait_closed()
+        await self.wait_closed()
 
         return self.collect_output()
     # pylint: enable=redefined-builtin
@@ -1482,20 +1513,6 @@ def __init__(self, process_factory: SSHServerProcessFactory,
         self._stdout: Optional[SSHWriter[AnyStr]] = None
         self._stderr: Optional[SSHWriter[AnyStr]] = None
 
-    async def __aenter__(self) -> 'SSHServerProcess[AnyStr]':
-        """Allow SSHProcess to be used as an async context manager"""
-
-        return self
-
-    async def __aexit__(self, _exc_type: Optional[Type[BaseException]],
-                        _exc_value: Optional[BaseException],
-                        _traceback: Optional[TracebackType]) -> bool:
-        """Wait for a full channel close when exiting the async context"""
-
-        self.close()
-        await self._chan.wait_closed()
-        return False
-
     def _start_process(self, stdin: SSHReader[AnyStr],
                        stdout: SSHWriter[AnyStr],
                        stderr: SSHWriter[AnyStr]) -> MaybeAwait[None]: