How to write ROS integration tests?

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That's is not an answer. Its a framework that has no examples on how to write a test

The first link under the Overview section is Writing rostests. Does that not provide sufficient information to at least get started?

No. It shows how to do gtest which is a unit test. An example of ROSTest(integration test) is not shown. Again the framework on how set set it up in the CMakefile.txt is shown but the actually writing of the test node is not.

The OP specifically asked:

For my test, I want to start roscore, several dependency nodes, my custom node, then publish messages, then wait N seconds for my node to publish its own messages, and fail if this doesn't happen.

rostest is how you do that. The rostest/Nodes page shows how to create .test files (essentially .launch files) that test what the OP appeared to ask for.

That's why I posted this answer.

Yes it does , but it does not tell you how to create the tester node. The tester node would have to publish messages to get the test started. I would assume the tester node would also have to subscribe to the result to determine that it occurred. There is no example on how to publish or subscribe and check for the result.

I don't believe that's how it works.

The "tester nodes" in the examples are already provided as part of the Standard Test Nodes (here).

The tester node would have to publish messages to get the test started.

That depends on what is getting tested: if the node-under-test is a subscriber, you'd have to write a small node that publishes messages, yes. Nothing special about that I believe.

If the NUT is a publisher, the provided standard test nodes could be used, at least as a starting point.

See rostest/nodes/publishtest as an example fi.

Now we are getting closer to an answer.

Using your above scenario if the node-under-test is a subscriber it would do some kind transformation/algorithm and output something. How would you know that the output is correct? In order fro an integration test to be valid you need to set up the test conditions and then verify that the results you are getting are correct. Having a test node publish the starting conditions is nothing special, but then you still have to verify the results.