Here's my recommended procedure. Do a Save as... with your current project (the one with the comps you want to copy), and give it a name like "My Project Reduced." Select only the comps you want to move into the new project, then click File > Reduce Project [link]. Save "My Project Reduced." This will remove all unselected comps and any unused media from the project copy.
Close that project and open the one you want to bring the comps into. File > Import, then choose "My Project Reduced."
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