Theoretically speaking, the statistical assumptions underlying Classical Test Theory say that the measures computed only make sense within the context of a particular assessment, so you should not try to combine results from a question being used in different tests (e.g. Descrimination index would be meaningless) or different cohorts (e.g. Facility index would be meaningless).
Practically speaking, some of the statistics, and particularly the response analysis, could be usefully aggregated across all tests.
See https://moodle.org/mod/forum/discuss.php?d=257037&mode=1 for more details and discussion.