While you can copy settings, synonyms and rules between indices, there is currently no out-of-the-box feature to auto-sync them.
However, when you modify/update an existing rule, you can select the option to copy the rule to other indices so that the indices will have the updated rule copied and replaced.
When you update the rule, you will need to manually add the indices you'd like to copy the rule to. Please note that this only applies to the manual editor.
