Throughout the bible, God makes it clear that the firstborn (of man and beast) and the firstfruits (of crops or produce) are set apart and belong to Him (Ex. 13:12, 22:29-30, Num. 18, Neh. 10:35-37, etc). Be Stirred Not Shaken 
But in reading the Old Testament, you see that God frequently elevated someone other than the firstborn for important roles. So by the time of the New Testament, it might be puzzling to read the emphasis on Jesus being the firstborn of many brethren (Rom 8:29).
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Old Testament examples where first born skipped over
Let's look at some of the places in the Old Testament where the firstborn is not chosen. How common was it? What patterns will emerge?
We only get four chapters into Genesis before we encounter conflict concerning the firstborn.
Gen 4:3 In the course of time Cain brought to the LORD an offering of the fruit of the ground, 4 and Abel also brought of the firstborn of his flock and of their fat portions. And the LORD had regard for Abel and his offering, 5 but for Cain and his offering he had no regard. So Cain was very angry, and his face fell.
We know how that turned out. I will return to this story later.
A few chapters later, we read the story of Noah and his sons after the flood. Normally, Noah's sons are given in the order “Shem, Ham, and Japheth”, as if they were triplets born in Noah's 500th year. But AFAICT that the actual birth order was Japheth, Shem, Ham (Gen 11:26 Gen 9:24) . It looks like Ham was pulling a power play, not only over his brothers, but over his father as well.
Sarah, Isaac, Abraham. Ishmael, Hagar
Or how about Esau and Jacob? Before the twins were born, God chose Jacob over Esau. Yet Rebekah and Jacob scheme to get the blessing and the birthright. Again, with hindsight, it looks like lack of faith.
The irony of Jacob’s life story is he spends his entire life manipulating and swindling his way into the blessing God already intended to give him. Bible Project 
I like Esau's reaction when meeting Jacob after 20 years. I wish Esau's descendants felt the same.
Gen 33:4 But Esau ran to meet him and embraced him and fell on his neck and kissed him, and they wept. ... 9 But Esau said, “I have enough, my brother; keep what you have for yourself.”
In Jacob's own family, his wife Leah was the firstborn but Jacob loved Rachel. This translated to envy among the sons of Leah against Joseph. Their envy really came out after Jacob gave Joseph a special coat, and after Joseph revealed his dreams. The blessing and the birthright did not go to Reuben, the first born because he disqualified himself. The birthright passed to Judah the fourth born, and the blessing went to Joseph, the 11th born - The Story About Dinah Is Not About Dinah .
And we're not even out of Genesis yet.
Moving to Exodus, let's look at an incident with Moses. He was not the firstborn, both Miriam and Aaron were older. Sibling rivalry came out long after childhood, they were all in their 80's.
Num 12:1 Miriam and Aaron spoke against Moses because of the Cushite woman whom he had married, for he had married a Cushite woman. 2 And they said, “Has the LORD indeed spoken only through Moses? Has he not spoken through us also?” And the LORD heard it.
The Lord heard it. and sorted them out immediately.
Let's skip forward a bit and look at two more examples, David and Solomon, then review what lessons we might draw.
God sent Samuel the prophet to anoint a king to replace Saul. In Bethlehem, Jesse brought seven sons before Samuel, but none of them would do. Samuel had to ask Jesse if he had any more sons. Then they all waited while someone fetched David the eighth son, whom Samuel anointed immediately. All of David's brothers saw the anointing by Samuel, but during the Goliath incident Eliab the firstborn still treated David as his baby brother.
1 Sam 17:28 Now Eliab his eldest brother heard when he spoke to the men. And Eliab’s anger was kindled against David, and he said, “Why have you come down? And with whom have you left those few sheep in the wilderness? I know your presumption and the evil of your heart, for you have come down to see the battle.”
Not a very respectful way to speak to an anointed king.
With Charles coronation, we saw the succession of kings go to the eldest son, this is known as primogeniture. Prince William is now first in line to the British throne. Prince George, first son of Prince William is second in line to the throne after his father. And so on, it's all mapped out - Independent . Firstborn son was and is the usual rule, but not so for David's successor Solomon. David had six sons with other wives before Bathsheba – Amnon, Absalom, and Adonijah are the most familiar.
Absalom followed his pride and greed and tried to seize his father’s throne. Learn Religions 
After the death of his elder brothers Amnon and Absalom, Adonijah considered himself the heir-apparent to the throne. Wikipedia 
King Charles, Princes George and William
Zero Sum Game
Let's consider the reaction of siblings that were “passed over”, that is firstborn who did not get the birthright. Most were envious. Absalom, and Adonijah tried to take over the throne. Esau threatened to kill Jacob. Jacob's sons sold Joseph into slavery. Cain murdered Abel.
The Cain and Abel story reveals things about envy, and it reveals God's perspective. Cain was upset that God did not regard his offering. We are not told exactly why. Cain engaged in what is now known as zero sum thinking.
Gen 4:6 The LORD said to Cain, “Why are you angry, and why has your face fallen? 7 If you do well, will you not be accepted? And if you do not do well, sin is crouching at the door. Its desire is for you, but you must rule over it.”
Despite this instruction from God, it appears Cain envied Abel and killed him. Perhaps he thought now God would have to bless him instead.
Being passed over as firstborn often led to envy.
We can examine ourselves for what we envy in others, maybe it's looks or a big house. But Envy has two sides. Some part of us likes envy, when we are the one being envied. It can make us feel good when someone envies OUR house or OUR fancy car. Whatever it is that you want others to envy is probably the thing that you envy in others. The Bible words for this
Envy is coveting
Wanting Envy is pride of life
We saw cases where the firstborn was passed over because they disqualified themselves. Or they didn't accept the responsibility that came with the birthright, or they didn't value it.
We saw cases where people tried to get the birthright as if they were entitled.
We saw cases where those passed over envied the one chosen – nearly all of them.
We see that Christ restored the rightful place of the firstborn.
We saw several cases where later born children were chosen then treated as firstborn. This pictures the church. We have no pedigree to claim firstborn status. I would argue that physically that would belong to the Jewish people. We aren't in line to succeed an earthly throne like King Charles or Prince William. We aren't the rich and powerful of this world. And we don't belong to a big influential Church. Yet all of these were passed over and WE were chosen. We believe God and obey Him because He chose us.
1 Pt 2:9 But you are a chosen race, a royal priesthood, a holy nation, a people for his own possession, that you may proclaim the excellencies of him who called you out of darkness into his marvelous light.
1 Co 1:27 But God chose what is foolish in the world to shame the wise; God chose what is weak in the world to shame the strong;
Today, WE are God’s firstfruits, spiritual Israel…those who have answered His calling, are keeping His commands, observing the sabbath and holy days, and striving to live a godly life (James 1:18, Rev. 14:4). 
May we accept the responsibility of the firstborn. May we value our calling over this world.