After reading the docs, I think you have to have to think differently about branch and channel.

Channels are specified at build time and exist inside a build's native code.

and

Branches are an ordered list of updates, similar to a Git branch, which is an ordered list of commits.

With EAS Update, we can link any channel to any branch, allowing us to make different updates available to different builds.

So first you have to "publish" the update group to a branch and then you gotta link the channel to point to that branch.

I imagine it could look like

eas update --branch staging-1.0.1
eas channel:edit staging --branch staging-1.0.1

Please correct me if I'm wrong about anything here.

https://github.com/expo/eas-cli#eas-channeledit-name

By default, expo-updates will target builds by Expo SDK version. It's kinda odd to be honest.

To answer your question, you would want to configure runtimeVersion.policy to nativeVersion. It is documented here.

You could set a specific version if you want but changing it to nativeVersion will let expo generate it for you. From the docs:

The runtime version associated with this manifest. Set this to {"policy": "nativeVersion"} to generate it automatically.

After setting it, your updates should now target that specific version+build automatically e.g. 1.0.1+69.

Example app.json

{
  "expo": {
    "runtimeVersion": {
      "policy": "nativeVersion"
    }
  }
}