Install Jenkins with both environment-script and parameterized-trigger plugins.

This WORKS:

Create a freestyle project.
Check "Generate environment variables from script"
Enter "echo Hello=World" into the script box.

Add a post-build action "Trigger parameterized build on other projects"
Select "Predefined parameters"
Enter "Hello=$Hello" into the params box.

Run the build. Note that the param to the next build is Hello=World.

This DOES NOT WORK:

Create a matrix project.
Check "Generate environment variables from script"
Enter "echo Hello=World" into the script box.
Check the "Run only on parent" box.

Add a post-build action "Trigger parameterized build on other projects"
Select "Predefined parameters"
Enter "Hello=$Hello" into the params box.

Run the build. Note that the param to the next build is Hello=$Hello.

Identical behavior if Run Only On Parent is checked or not, but I expect that this is a required option, because the post-build steps occur on the parent as well.