Trigger transactional mailer via Custom Object

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Hi,

We have a use case whereby we are looking to trigger a transactional mailer via a custom object, where the object is being updated via the REST API. The object contains a field which can have one of 4 values, or 'triggers', each of which corresponds to a separate mailer. More than one trigger can be hit at once; leading to other actions being taken, such as an incident being created or another field being populated. 

I have been trying to trigger the mailers via Enhanced Business Rules, however it seems that I can only create rules that act on the custom object and not in the contacts or incidents table for example.