Drupal 10.4 installation does not recognise EVA updates

Well, you should use composer to find out how it resolved dependencies. Helpful commands are composer why drupal/eca and composer why-not drupal/eca 2.1.0.

Thanks for the response.

When I issue that command, I see...

$ composer why drupal/eca
drupal/recommended-project dev-main requires drupal/eca (^2.1)
drupal/bpmn_io             2.0.4    requires drupal/eca (^2.0)
drupal/eca_flag            2.0.1    requires drupal/eca (^2.0)
drupal/eca_modeller_bpmn   2.1.0    requires drupal/eca (^2)
drupal/eca_tamper          2.0.2    requires drupal/eca (^2.0)
drupal/eca_ui              2.1.0    requires drupal/eca (*)

I'm not sure this is necessarily to do with dependencies is it - from the UI screenshot the problem appears to be communication with wherever it communicates to be made aware of the update(s)?

Issuing the command...

composer why-not drupal/eca 2.1.0

Reveals...

There is no installed package depending on "drupal/eca" in versions not matching 2.1.0

Well, your issue says that composer update is not giving you ECA updates. But it looks like it does. You do have the latest ECA version installed.

The second issue is about the Drupal core update module, which can't receive update information from drupal.org, which is totally unrelated to composer. In the issue queue of Drupal core there are a ton of issues about that phenomenon, and it's about local network, DNS resolution and what not. It's nothing to do with ECA or any of your other 5 modules. It's only about the update module not being able to receive the necessary data.

OK, got it, thank you.
That's very useful indeed.

It turned out to be rogue remnants in the key_value table...

That would explain the other five offending modules too!