Error "insufficient access rights on cross-reference id" on a Case record

This error can occur when the user doesn't have access to the email template folder used by the process.

To resolve the issue, we need to share the folder to the user profile by following the steps below:

In Salesforce Classic: Setup | Communication Templates | Classic Email Templates
In Lightning Experience: Gear Icon | Setup | Email | Classic Email Templates
 

Perform the following:

1. Click the folder dropdown and locate the folder of the email template. 
   Note: Not all folders will be available to edit. For example, the Unified Public Classic Email Templates is an out-of-the-box template that cannot be edited.
   
2. Click Edit.
3. Select This folder is accessible by all users or This folder is accessible only by the following users: and select the users through Public Groups, Roles, and Territories grouping. Move the group from Available for Sharing to Shared To.
4. Click Save.

Notes:

• The Folders and Enhanced Sharing option is available in Salesforce Lightning to allow users to create, edit, and delete folders and share their folders and templates.  In Setup, select Email | Lightning Email Templates and switch the setting to On.
• For more information on Managing Folders please review this Salesforce Help.
• If receiving this error sending emails, please review the steps listed in the Salesforce article, Error "Insufficient access rights on Cross Reference ID" on Attempts to Send Emails.
  Please note that Case Assignment Rules are invoked immediately upon saving the Case record, as part of the standard order of execution. The error "Insufficient access rights on cross-reference ID" can occur if the user does not have access to the Email Template Folder associated with the template used in the Case Assignment Rule. Hence this behavior should be considered when troubleshooting access-related issues during automated operations like email notifications.
• View the information regarding this error and the API in API Error 'insufficient access rights on cross-reference id'.

Because this error has several causes, if the above information does not assist, please contact Salesforce Support.