It's well known that 12 tribals from Israel come from Jacob (Abraham's grandson), that means that Israel was already free and in place. However it's not clear the events that connect these two characters during the time.
The fact that Moses was a physical descendant of Abraham according to the flesh is clearly illustrated in the genealogical charts within the Scripture. Moses was a brother of Levi, but not a Levite himself. Moses acted outside of the Levitical priesthood, even though he performed the role of a priest when he mediated on behalf of Israel before God. Priests are mediators between God and man.
Moses was a prophet of God and a mediator between man and God
“The Lord your God will raise up for you a prophet like me from among you, from your brethren—him you shall heed (Deuteronomy 18:15)
And the Lord said to Moses, “How long will this people despise me? And how long will they not believe in me, in spite of all the signs which I have wrought among them?I will strike them with the pestilence and disinherit them, and I will make of you a nation greater and mightier than they.”
But Moses said to the Lord....Pardon the iniquity of this people, I pray thee, according to the greatness of thy steadfast love, and according as thou hast forgiven this people, from Egypt even until now.” Then the Lord said, “I have pardoned, according to your word. (Exodus 14:11-20)
Abraham was a prophet of God and a mediator between God and man.
Then God said to him in the dream, “Yes, I know that you have done this in the integrity of your heart, and it was I who kept you from sinning against me; therefore I did not let you touch her. Now then restore the man’s wife; for he is a prophet, and he will pray for you, and you shall live. But if you do not restore her, know that you shall surely die, you, and all that are yours.” So Abimelech rose early in the morning, and called all his servants, and told them all these things; and the men were very much afraid. Then Abimelech called Abraham, and said to him.... (Genesis 20:6-9)
Moses possessed the faith of Abraham
What is important about Moses is that he possessed faith. Many people do not possess faith. They confuse faith with anything but what it is, really. Morons say faith is a mere wish. Imbeciles have no idea. Idiots stumble in the dark with their definitions. But faith is the inherent ability to act upon knowledge because greater things are foreseen. Faith is not a body of knowledge that people believe such as scientific theories. Faith is demonstrated by action. Although, many people claiming to be Christians think faith is their set of doctrines that they profess which distinguishes them from the people claiming to be Christians in the building down the road, or in some cases, next door.
The Bible teaches that Abraham is the father of all who believe and he is the example of faith for all to follow.
Abraham received circumcision as a sign or seal of the righteousness which he had by faith while he was still uncircumcised.The purpose was to make him the father of all who believe without being circumcised and who thus have righteousness reckoned to them, and likewise the father of the circumcised who are not merely circumcised but also follow the example of the faith which our father Abraham hadbefore he was circumcised. (Romans 4:11-12)
Abraham is the father of all who believe and follow his example of faith, whether they are circumcised or uncircumcised. For it is only through faith that people are justified before God. It is only through faith that God honors those who are justified. This is where the true connection between Moses and Abraham lies. Not only was Moses, Abraham’s seed according to the flesh, but he was also his seed according to faith. For we read
By faith Moses, when he was born, was hid for three months by his parents, because they saw that the child was beautiful; and they were not afraid of the king’s edict.
By faith Moses, when he was grown up, refused to be called the son of Pharaoh’s daughter, choosing rather to share ill-treatment with the people of God than to enjoy the fleeting pleasures of sin. He considered abuse suffered for the Christ greater wealth than the treasures of Egypt, for he looked to the reward.
By faith he left Egypt, not being afraid of the anger of the king; for he endured as seeing him who is invisible.
By faith he kept the Passover and sprinkled the blood, so that the Destroyer of the first-born might not touch them. By faith the people crossed the Red Sea as if on dry land; but the Egyptians, when they attempted to do the same, were drowned. (Hebrews 11:23-29 RSV)
The connection between Abraham and Moses is twofold. One is by natural descent. The other is because Moses demonstrated that he possessed the faith of Abraham. This faith is essential for all people to demonstrate if they are to find favor in the eyes of the Lord God. Consequently, we see both Abraham and Moses being declared prophets and successfully mediating on behalf of other people. For without faith you cannot do anything. With faith all things are possible.