What happens to kids born/growing up after the Rapture?

Welcome to Unsealed rg86 - you've asked quite the loaded question with there being a few differing views on the timing of the rapture and length of the Tribulation. However from the view that you present, I think Yardstick has given some valid guidance. Those that would remain after a (7 year) pre-trib rapture would all effectively be considered non-believers. A step further in my mind is a God-hater, one who has no regard for our Creator, His ways, guidance, rules, etc. (Rom 1:30, Jam 4:4, among others). Taking a look back where those societies have no regard for Him we see things like sacrificing children to Molech, we can see how the Lord views the people who commit such heinous acts

Lev 20:2 Again, thou shalt say to the children of Israel, Whosoever he be of the children of Israel, or of the strangers that sojourn in Israel, that giveth any of his seed unto Molech; he shall surely be put to death: the people of the land shall stone him with stones. 3 And I will set my face against that man, and will cut him off from among his people; because he hath given of his seed unto Molech, to defile my sanctuary, and to profane my holy name. 4 And if the people of the land do any ways hide their eyes from the man, when he giveth of his seed unto Molech, and kill him not: 5 Then I will set my face against that man, and against his family, and will cut him off, and all that go a whoring after him, to commit whoredom with Molech, from among their people.

As you are likely aware, if the last 7 years before the Second Coming of the Messiah are to look like we think they may, the children have much to be concerned with physically. They will be left in a virtually Godless world with tyrannical leaders LIKE Gov A Cuomo, who do not hold value on innocent lives. BUT GOD! He is merciful and though we may not understand how He would or could allow the types of pagan practices of old to happen when they did, He in fact allowed it. 

EDIT - something I personally wrestle with is where I fix my vision. When I think on things that are earthly I get earthly results. But when the Holy Spirit guides me beyond the natural sight I see that though children in the Tribulation may become an innocent victim of horrible consequence they are in fact innocent and god is merciful. Their physical bodies not being the reward, rather the eternal ones are what He shows me.

I speculate about this sometimes too. I know in the end the Lord is enduring and with children, well they are innocent to a degree. I dont want to tangent your question here rg86 (as I notoriously do sometimes) but I do also consider this:
 
Mark 9:42 And whosoever shall offend one of these little ones that believe in me, it is better for him that a millstone were hanged about his neck, and he were cast into the sea.
Luke 17:2 It were better for him that a millstone were hanged about his neck, and he cast into the sea, than that he should offend one of these little ones.

Jesus certainly warns the teachers of little ones here but notice "believes in me" - I try to teach my kids about the Lord and I trust they believe. But what about those who teach their children contrary? Just a food for thought kinda question, that doesnt really need a reply

The other piece I consider is the tribulation and the way that will actually look to a child. The godzilla bros. (stealing that from disciple4life) on the scene may look like the Marvel or DC heroes they have come accustomed to. Perhaps they would be more apt to listen to them or maybe this Rev 14:6-7 would have an impact

I suppose that it is like the days of Noah; after the rapture then the earth is repopulated with families: children, etc.

ie. Noah's flood as Noah's rapture wherein Noah and his family were taken from the earth upon the water and similarly we are taken from the earth upon the Living Water (Christ). In both cases the people are returned to the earth. Please notice in Rev. 21 that the holy city comes to the earth as does Christ in Rev. 22. Naturally, the people come with the city and Christ. 

venge,

Thanks so much to everyone who replied. Venge, thank you for your thoughts too. I'd like to hear more about what you mean.

What I have read on this website and elsewhere, and my understanding from the Scriptures, is a pre-trib rapture followed eventually by a 7-year period of judgments, anti-Christ, and so forth. 7 years split by 3.5 year increments of varying degrees of severity and events, third temple, abomination of desolation, etc etc. In this view, all the pre-tribulation saints would be raptured, and would not endure God's wrath. However, people would turn to the Lord in the middle of the 7 year tribulation, after the rapture, and thus become what is usually called "tribulation saints." That's always how I've heard this chronology described. Those tribulation saints by and large will be martyred.

If I correctly understand, you are saying that you think this is incorrect, and that there is a shorter period of time, not 7 years. Are you referring to what I've heard described as the "Pre-Wrath/Mid Trib" position? Or are you referring to something else? I'd love to learn more about what the chronology you're thinking about is. Forgive me if I've somehow misunderstood your position. I'm hoping you'll elaborate.

Thanks!