No. If you are talking about the Child Tax Credit, yes, you have to have earned income in order to claim it so if you took the foreign income exclusion, you would not be able to claim it.
The Child Tax Credit is a non-refundable credit that just lowers your tax liability. Then there is the Additional Child Tax Credit which is worth up to $1,700 per child. This again, does require that you have earned income.
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