Thanks for your posting. Test results/runs cannot be moved/copied to different projects or test suites currently. This is because a test run and the related test results are always associated with a specific test suite and it may not make sense to transfer test results from one test run to another in all cases (as they may run in different contexts or use different configurations, for example).
Before restructuring your test suites, you would usually close old test runs to archive them and then just create new test runs to manage your tests with the new structure. Closing the test runs is an important step so that your existing test history is archived and can still be used to access old test results.
Just let me know in case you have any further questions.