Bump process_runner from 4.2.0 to 4.2.2 in /app_dart (#4865)

Bumps [process_runner](https://github.com/google/process_runner) from 4.2.0 to 4.2.2.
<details>
<summary>Changelog</summary>
<p><em>Sourced from <a href="https://github.com/google/process_runner/blob/main/CHANGELOG.md">process_runner's changelog</a>.</em></p>
<blockquote>
<h2>4.2.2</h2>
<ul>
<li>Refactoring the main <code>run</code> method in <code>ProcessRunner</code> into smaller, more
manageable private methods.</li>
<li>Enhancing <code>FakeProcessManager</code> to differentiate commands based on their
working directory and the <code>runInShell</code> parameter, and adding more
comprehensive tests for it.</li>
<li>Making the addition of dependencies in <code>ProcessPool</code> idempotent (do nothing if
the dependency already exists).</li>
<li>Updating the CI to track the main branch.</li>
<li>Adding a number of new tests to cover <code>stdin</code> handling, <code>runInShell</code> behavior,
and exception scenarios.</li>
</ul>
<h2>4.2.1</h2>
<ul>
<li>Updates analysis_options.yaml to modern lints, and updates formatting.</li>
<li>Moves <code>FakeProcessManager</code> to the main package so developers can use it for
testing. Not exported from the main package import.</li>
</ul>
</blockquote>
</details>
<details>
<summary>Commits</summary>
<ul>
<li>See full diff in <a href="https://github.com/google/process_runner/commits">compare view</a></li>
</ul>
</details>
<br />

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</details>

Author: dependabot[bot]

app_dart/pubspec.yaml

@@ -42,7 +42,7 @@ dependencies:
   neat_cache: 2.0.4
   path: ^1.9.1
   process: ^5.0.3
-  process_runner: 4.2.0
+  process_runner: 4.2.2
   protobuf: 3.1.0
   retry: ^3.1.2
   truncate: 3.0.1