The blood of the covenant in the Old and New Testaments

Exodus 24:

8 Moses then took the blood, sprinkled it on the people and said, “This is the blood of the covenant that the LORD has made with you in accordance with all these words.”

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Matthew 26:

27 [Jesus] took a cup, and when he had given thanks he gave it to them, saying, “Drink of it, all of you, 28 This is My blood of the covenant, which is poured out for many for the forgiveness of sins.

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In the OT, Moses sprinkled the blood on the people. In the NT, we symbolically drink the blood to get it inside us.

Hebrews 12:

24 to Jesus the mediator of a new covenant, and to the sprinkled blood that speaks a better word than the blood of Abel

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i.e., blood that cried out to God.

Three chapters earlier in Hebrews 9:

18 Therefore not even the first covenant was inaugurated without blood. 19For when every commandment of the law had been declared by Moses to all the people, he took the blood of calves and goats, with water and scarlet wool and hyssop, and sprinkled both the book itself and all the people, 20saying, “This is the blood of the covenant that God commanded for you.” 21 And in the same way he sprinkled with the blood both the tent and all the vessels used in worship. 22 Indeed, under the law almost everything is purified with blood, and without the shedding of blood there is no forgiveness of sins.

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This described a type of Christ's sacrifice and blood sprinkling. They used dried hyssop as a handle and wool to soak up the blood for sprinkling. The true type was described in:

11 But when Christ appeared as a high priest of the good things that have come, then through the greater and more perfect tent (not made with hands, that is, not of this creation)

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Instead of a human high priest, the true type has Christ as the high priest.

12 he entered once for all into the holy places, not by means of the blood of goats and calves but by means of his own blood, thus securing an eternal redemption.

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Instead of animal blood, the true type is the blood of Christ.

13 For if the blood of goats and bulls, and the sprinkling of defiled persons with the ashes of a heifer, sanctify for the purification of the flesh, 14 how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God, purify our conscience from dead works to serve the living God.

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This blood of the new covenant gets inside us to purify our conscience to serve God.