Testing in Gitlab CI/CD with different dependacies versions

I'm currently developing a (Laravel) package on Gitlab, and i want to automate testing using its CI/CD pipeline.

The problem

I already know ho to set up a pipeline in Gitlab, but what i want to achieve is to automate testing against different versions of the same dependancy, in order to keep checking compatibility with old version and add checking with upcoming new ones.

The case

My Laravel package is not so complex right now and don't use some particular nor specific Laravel features, so i would like to keep it compatible with te more versions of laravel possible: i would trigger different testing stages in my pipeline to run my tests against laravel 5.6, 5.7, 5.8, 6, 7, and 8.

The question

How do i trigger different testing stages using different laravel/framework versions?
When downloading dependancies composer will go for the latest version available if i define it with '^', so which files do i have to edit?