This should be a relatively simple request I think, but I can't find another example that has the specifics I need.

I would like to update one object based on changes made to another object where one string is equivalent.  In this case, I have a custom object (Customobj__c) that has a trigger on it to update a case with fields from that custom object based on a field that they both have in common.  One of the fields I want to update is a multi-select picklist, too (Event_Types__c).  The primary weirdness I am encountering is that the multi-select picklist doesn't always update when I save the Custom Object record, but I also suspect my code isn't laid out efficiently.  Can anyone help?

Here's the code:

trigger OnCustobjCaseFields on Customobj__c (before insert, before update)
{
    String objSharedField;
    String objRecordId;
    String objAccount;
    String objEventTypes;
    
    for(Customobj__c obj: Trigger.new)
    {
        if (obj.Shared_Field__c != Null)
        {
            objSharedField = obj.Shared_Field__c;
            objRecordId = obj.Id;
            objAccount = obj.Account__c;
            objEventTypes = obj.Event_Types__c;
        }
    }
    
    List<Case> cases = new List<Case>([SELECT Id, Shared_Field__c, AccountId, Expected_Event_Types__c FROM Case WHERE Shared_Field__c = :objSharedField]);
    List<Case> caseUpdateList = new List<Case>();
    
    for(Case c: cases)
    {
        c.AccountId = objAccount;
        c.App__c = objRecordId;
        c.Expected_Event_Types__c = objEventTypes;
        caseUpdateList.add(c);
    }
    
    if(caseUpdateList.size() > 0)
    {
        update caseUpdateList;
    }
}