In Volume 3 of his magisterial Reformed Dogmatics, Herman Bavinck has a wonderful passage on the continuity between the Abrahamic and Mosaic covenants. There are several important things to note.
First, Bavinck says that the covenant with Israel is based on God’s promises and rooted in God’s grace no less than the covenant with Abraham:
The covenant with Israel was essentially no other than that with Abraham. Just as God first freely and graciously gave himself as shield and reward to Abraham, apart from any merits of his, to be a God to him and his descendants after him, and on that basis called Abraham to a blameless walk before his face, so also it is God who chose the people of Israel, saved it out of Egypt, united himself with that people, and obligated it to be holy and his own people. The covenant of Mount Sinai is and remains a covenant of grace. “I am the Lord your God, who brought you out of the land of Egypt, out of the house of slavery” (Exod. 20:2) is the opening statement and foundation of the law, the essence of the covenant of grace. …
The benefits granted to Israel by God in this covenant are the same as those granted to Abraham, but more detailed and specialized. … The one great promise to Abraham is “I will be your GOd, and you and your descendants will be my people” (Gen. 17:8 paraphrase). And this is the principal content of God’s covenant with Israel as well. God is Israel’s God, and Israel is his people (Exod. 19:6; 29:46; etc.).
Second, the obligations of the covenant with Israel are the same as the obligations of the covenant with Abraham, summarized in Genesis 17:1 – “Walk before me and be blameless.”
Just as Abraham, when God allied himself with him, was obligated to “walk before his face,” so Israel as a people was similarly admonished by God’s covenant to a new obedience. The entire law, which the covenant of grace at Mount Sinai took into its service, is intended to prompt Israel as a people to “walk” in the way of the covenant. It is but an explication of the one statement to Abraham: “Walk before me and be blameless” (Gen. 17:1), and therefore no more a cancellation of the covenant of grace and the foundation of a covenant of works than this word spoken to Abraham. The law of Moses, accordingly, is not antithetical to grace but subservient to it and was also thus understood and praised in every age by pious men and women.
Third, Bavinck says that Paul’s polemic against “the law” is in part because of the misuse of the law by Paul’s opponents:
But detached from the covenant of grace, it indeed became a letter that kills, a ministry of condemnation.
Fourth, Bavinck says that the argument of the New Testament regarding the Mosaic covenant is redemptive-historical: Jesus is the fulfillment of the law, the one to whom the law pointed all along, the one apart from whom there is no salvation:
Another reason why in the time of the Old Testament the covenant of grace took the law into its service was that it might arouse consciousness of sin, increase the felt need for salvation, and reinforce expectation of an even richer revelation of God’s grace. It is from that perspective that Paul views especially the Old Testament dispensation of the covenant of grace. He writes that Israel as a minor, placed under the care of the law, had to be led to Christ (Rom. 10:4; Gal. 3:23f.; 4:1f) and that in connection sin would be increased and the uselessness of works for justification and the necessity of faith would be understood (Rom. 4:15; 5:20; 7:7f; 8:3; Gal. 3:19).
Herman Bavinck, Reformed Dogmatics [Grand Rapids: Baker Academic, 2006], 3.220-222.
Compare the later work of John Murray, which also gets this issue pretty much exactly right: