Gitslave is used when you want to run the same command on multiple repos. The only applicability in this case would be if you created a superproject (potentially out of basic repo, possibly another repo) and had all of the project repos as children. You could then, in one command, add a remote in all of the slave repos pointing at the basic repo, check out the basic-branch in all slave repos, then pull the changes into each slave repo, and finally merge/rebase the basic-branch onto the project branch (master or whatever). Once you have resolved any conflicts, you can then push each project repo out.
This is the inverse of pushing the changes into the other repositories, but probably much more useful since you can then automate the merge process as well.
Note another approach which is more consistent of your original desire would be to add the "all" target for push which would push the basic-branch to multiple repos in the same command. http://jeetworks.org/node/22
However, please note that you must have a "basic-branch" concept under all circumstances to have a sane branching strategy. Specifically, the project branch must be branched from basic-branch so that updates to basic-branch can be merged or rebased in.
Still another option would be to have one repo to rule them all, and create one branch per project. You would then need to write some automation to merge your changes to basic-branch into all other appropriate branches. Depending on your needs, security, or how far the projects drift from each other, or how much project-specific cruft there is, this may or may not make sense.