I have 3 children who received 1099's from their UTMA brokerage custodial accounts for various amounts of income and gains. Do I need to file separate returns for them?

With the information in your question, two of your children don’t have to file returns.  Among other factors, their investment income would have to be more than $1050.00. 

Your other child will need to file a return.  You could report the $1700 on your own return.  In the search box in the upper right corner of your TurboTax return, type in child income and click on the magnifying glass.  You will receive a Jump To link.  Please click on the link. (See screenshot below)

Please go to the links below for further information.

https://ttlc.intuit.com/questions/1901402-do-i-need-to-file-my-own-taxes-if-i-m-a-dependent