Is it possible to create a row-action button to push contact field to custom object field

We have a phone field on the contact level, and would like to push it to our custom object “family” level by clicking a button or checking a box. Workflow rules don’t help us here because cross-object workflow rules don’t work for custom objects. There is a one-to-many relationship with families and contacts, so there will always only be one related family to a contact. This is critical functionality for our org, so please let us know of any possible solutions.