Praise be to Allah.
The destiny of the children of believers is Paradise, because they follow their parents. Allaah says (interpretation of the needed):
“And those who believe and whose offspring follow them in Faith, — to them shall We join their offspring, and We shall not decrease the reward of their deeds in anything. Every person is a pledge for that which he has earned”
As for the children of non-believers, i.e., children born to non-Muslim parents, the most correct view is that Allaah knows best what they would have done, so in this world they come under the same rulings as their parents; with regard to the Hereafter, Allaah knows best what they would have done, as the Prophet (blessings and peace of Allaah be upon him) said, and Allaah knows best about their fate. This is what we say, and in fact this is something that does not greatly concern us. Rather what concerns us is the ruling concerning them in this world. The ruling concerning them -- meaning the children of the mushrikeen -- is that they are like the mushrikeen: (if they die) they are not to be washed or shrouded, the funeral prayer is not to be offered for them and they are not to be buried in the Muslim graveyard. And Allaah knows best. End quote.