Filter multiple post types with single taxonomy

I need a custom shortcode (could be in a plugin or just the code) which will display a grid layout of various custom post types filtered by the same taxonomy. So, say for example, that the custom post types are post, cpt1, cpt2, cpt3 where post, cpt1 and cpt2 share the same taxonomy (the built-in category taxonomy). The shortcode should display the category filter as a drop-down and 3 latest posts from each post type (not mixed up but separately with their CPT title above each grid of 3). When user changes the category from the drop-down the results should change accordingly. For cpt3 which does not have a taxonomy results should remain the same on filter change.