Thunderbird can only automatically forward incoming messages. Log into your child's account and click Tools>>Message filters. Click the New button.
Give the filter a title in the “Filter name” field. Select “Match any of the following.” In the first drop-down menu, select From. For the second, select “is in my address book.” Ignore the third, and click the + button. A new line of drop-down menus will appear. In the second new menu, select “isn’t in my address book.”
Move down to the “Perform these actions” field. In the first drop-down menu, select “Forward Message to.” Then, type your email address in the blank field. Click OK.
Don’t forget to use both programs’ built-in spam and content filters. You can also try add-on spam filters. Some choices are MailWasher or SpamBayes.