`TypeError: cannot pickle '_thread.RLock' object` with `ProcessPoolExecutor`

When Trying to use ProcessPoolExecutor, I get an error:

code/pydantic-ai 130 ➤ uv run pytest tests/test_examples.py -k graph --update-examples     
============================================================================================= test session starts ==============================================================================================
platform darwin -- Python 3.12.6, pytest-8.3.3, pluggy-1.5.0
rootdir: /Users/samuel/code/pydantic-ai
configfile: pyproject.toml
plugins: inline-snapshot-0.14.0, logfire-2.6.2, devtools-0.12.2, examples-0.0.15, mock-3.14.0, pretty-1.2.0, anyio-4.6.2.post1
collected 118 items / 101 deselected / 17 selected                                                                                                                                                             

tests/test_examples.py ..........F                                                                                                                                                                       [ 64%]
tests/test_examples.py:55 test_docs_examples[docs/graph.md:643-709] - TypeError: cannot pickle '_thread.RLock' object…
tests/test_examples.py s.....                                                                                                                                                                            [100%]

=================================================================================================== FAILURES ===================================================================================================
__________________________________________________________________________________ test_docs_examples[docs/graph.md:643-709] ___________________________________________________________________________________

example = CodeExample(source='from __future__ import annotations\n\nimport asyncio\nfrom concurrent.futures.process import Proce...D('99e16cec-fcc7-44ff-973e-449f8e7bc3d8'), test_id='tests/test_examples.py::test_docs_examples[docs/graph.md:643-709]')
eval_example = <pytest_examples.eval_example.EvalExample object at 0x109d66c00>, mocker = <pytest_mock.plugin.MockerFixture object at 0x109d66ae0>
client_with_handler = <async_generator object client_with_handler at 0x109d02180>, env = <tests.conftest.TestEnv object at 0x109d668d0>
tmp_path = PosixPath('/private/var/folders/s6/mh0x1z6d6mqclzdtv72zgx_40000gn/T/pytest-of-samuel/pytest-782/test_docs_examples_docs_graph_10'), set_event_loop = None

    @pytest.mark.parametrize('example', find_filter_examples(), ids=str)
    def test_docs_examples(
        example: CodeExample,
        eval_example: EvalExample,
        mocker: MockerFixture,
        client_with_handler: ClientWithHandler,
        env: TestEnv,
        tmp_path: Path,
        set_event_loop: None,
    ):
        mocker.patch('pydantic_ai.agent.models.infer_model', side_effect=mock_infer_model)
        mocker.patch('pydantic_ai._utils.group_by_temporal', side_effect=mock_group_by_temporal)
        mocker.patch('pydantic_ai.models.vertexai._creds_from_file', return_value=MockCredentials())
    
        mocker.patch('httpx.Client.get', side_effect=http_request)
        mocker.patch('httpx.Client.post', side_effect=http_request)
        mocker.patch('httpx.AsyncClient.get', side_effect=async_http_request)
        mocker.patch('httpx.AsyncClient.post', side_effect=async_http_request)
        mocker.patch('random.randint', return_value=4)
        mocker.patch('rich.prompt.Prompt.ask', side_effect=rich_prompt_ask)
    
        env.set('OPENAI_API_KEY', 'testing')
        env.set('GEMINI_API_KEY', 'testing')
        env.set('GROQ_API_KEY', 'testing')
    
        sys.path.append('tests/example_modules')
    
        prefix_settings = example.prefix_settings()
        opt_title = prefix_settings.get('title')
        opt_test = prefix_settings.get('test', '')
        opt_lint = prefix_settings.get('lint', '')
        noqa = prefix_settings.get('noqa', '')
        python_version = prefix_settings.get('py', None)
    
        if python_version:
            python_version_info = tuple(int(v) for v in python_version.split('.'))
            if sys.version_info < python_version_info:
                pytest.skip(f'Python version {python_version} required')
    
        cwd = Path.cwd()
    
        if opt_test.startswith('skip') and opt_lint.startswith('skip'):
            pytest.skip('both running code and lint skipped')
    
        if opt_title == 'sql_app_evals.py':
            os.chdir(tmp_path)
            examples = [{'request': f'sql prompt {i}', 'sql': f'SELECT {i}'} for i in range(15)]
            with (tmp_path / 'examples.json').open('w') as f:
                json.dump(examples, f)
    
        ruff_ignore: list[str] = ['D', 'Q001']
        # `from bank_database import DatabaseConn` wrongly sorted in imports
        # waiting for https://github.com/pydantic/pytest-examples/issues/43
        # and https://github.com/pydantic/pytest-examples/issues/46
        if 'import DatabaseConn' in example.source:
            ruff_ignore.append('I001')
    
        if noqa:
            ruff_ignore.extend(noqa.upper().split())
    
        line_length = int(prefix_settings.get('line_length', '88'))
    
        eval_example.set_config(ruff_ignore=ruff_ignore, target_version='py39', line_length=line_length)
        eval_example.print_callback = print_callback
    
        call_name = prefix_settings.get('call_name', 'main')
    
        if not opt_lint.startswith('skip'):
            if eval_example.update_examples:  # pragma: no cover
                eval_example.format(example)
            else:
                eval_example.lint(example)
    
        if opt_test.startswith('skip'):
            print(opt_test[4:].lstrip(' -') or 'running code skipped')
        else:
            if eval_example.update_examples:  # pragma: no cover
>               module_dict = eval_example.run_print_update(example, call=call_name)

tests/test_examples.py:133: 
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 
docs/graph.md:694: in main
    result, history = await fives_graph.run(DivisibleBy5(3), deps=deps)
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 

    async def run(self, ctx: GraphContext) -> DivisibleBy5:
        loop = asyncio.get_running_loop()
>       compute_result = await loop.run_in_executor(
E       TypeError: cannot pickle '_thread.RLock' object

docs/graph.md:678: TypeError
=============================================================================================== inline snapshot ================================================================================================
                                                   Summary of Failures                                                   
┏━━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━┳━━━━━━━━━━━━━┓
┃  File                    ┃  Function                                   ┃  Function Line  ┃  Error Line  ┃  Error      ┃
┡━━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━╇━━━━━━━━━━━━━┩
│  tests/test_examples.py  │  test_docs_examples[docs/graph.md:643-709]  │  56             │  133         │  TypeError  │
└──────────────────────────┴─────────────────────────────────────────────┴─────────────────┴──────────────┴─────────────┘
Results (0.79s):
         1 failed
        15 passed
         1 skipped
       101 deselected

Ref https://github.com/pydantic/pydantic-ai/pull/693

Reproduce the failure with uv run pytest tests/test_examples.py -k graph --update-examples, then inspect docs/graph.md lines 643-709 and tests/test_examples.py around lines 55 and 133. The work is done when the graph example completes without the pickle TypeError and the example test passes.