Within the system a household member can only be associated with one household at a time. Within any client's [Relationships] tab you'll be able to see who is "Living With..." or simply "Related To..." to the client. For example, If John Smith lives with 4 people and 1 other person is his child, Tommy Smith, who lives with his mother at another address... you would see Tommy Smith listed under "Related To..." (see screenshot here). The [Relationships] tab will offer a helpful overview of all dynamics of relationships.
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A possible solution could be to create an [Alert] on the client's file to identify that that a child is being claimed at more than one household (see screenshot here). The "Alert" would pop-up first and foremost when the Case file is opened in the future to provide a heads-up!