We had been developing exactly that, one Omeka for multiple museums (120) each one with multiple collections and multiple types of elements.
What worked for us:
- one collection named "Museums" where the items have a code (obtained from institutional organization chart) in the DC field 'Identifier'.
- multiple collections, corresponding the different collections of each museum, Those collection have their museum's code in the DC field 'Relation'.
-four big collections: "Exhibits", "Events", "Publications" and "Spaces". Those collections contain the elements of the proper Item type, for all museums, each one with their museum code in the DC 'Relation'.
This way we can have a Home that displays pictures of all the museums in a slider, linked to the home of that museum. also, four small sliders that show, randomized, all Exhibits, Events, Publications and Spaces, each one with it's title and a link to their museum.
We wrote a "SuperFilterPlugin" that recibe all links and filters the requests by Item type and by the appropriate DC field (for the appropriate museum), builds an URL to a Simple Page named "Superpage", appending the code of the museum and it's Id from the 'Items' table, to the URL,
Then, "Superpage" receives the URL with the variables via GET, and with the help of a custom 'browse.php', displays the appropriate items.
Hope that helps, I can share more info about this work if it is helpful for someone.