What are the covenant promises based on?

All the covenant promises are based on God's promise of the coming of the seed of the woman, which is Christ. That promise, first given in Genesis 3:15, runs through the Jewish patriarchs and the nation of Israel, and it is the foundation for the four programs mentioned.

Les explains that the covenant promises—the whole four programs he has been talking about—are all based on God's promise of the coming of the seed of the woman, which is Christ. He shows that the seed of the woman in Genesis 3:15 comes right on up through the Jewish patriarchs and the little nation of Israel. The same promise is concerning His Son, Jesus Christ our Lord, who was made of the seed of David according to the flesh, and declared to be the Son of God with power.

Genesis 3:15

And I will put enmity between thee and the woman, and between thy seed and her seed; it shall bruise thy head, and thou shalt bruise his heel.

How does the Abrahamic covenant connect to the promise of a Redeemer in Genesis 3?

The Abrahamic covenant is built upon Genesis 3:15, where God promised a Redeemer through the woman. That promise is then narrowed to Abraham's offspring, through whom the seed of the woman, Christ, would come, and through Abraham all families of the earth would be blessed.