Richard J. Clifford points out some of the similarities between the OT experience of Israel in the Exodus and Wilderness, with the experience of believers in Christ in the NT. He writes:

In the Gospels the Exodus appears in the voice of John the Baptist crying in the wilderness and the theme of the Way of the Lord. Matthew and Luke interpret the temptations of Jesus in the light of the Exodus: 40 days in the desert recalls Israel's 40 years; as Israel was guided by the column of fire so Jesus is guided by the spirit; Jesus unlike Israel does not succumb to temptation; Israel rebelled over food, Jesus subordinates food to the word of God; Israel demanded signs, Jesus refuses to tempt God; rather than worshiping a thing (the golden calf) Jesus declares God alone worthy of adoration. He is the perfect realization of Israel. Paul considers Jesus as the paschal victim immolated for us (1 Corinthians 5:7); in the wonders of the Exodus, he discovers the spiritual realities represented by Christ (! Corinthians 10:1-6). 1 Peter uses images from the Exodus to explain the Church: the blood of the lamb, the call of Christians, light, the pagan life that one must abandon as Israel left idolatrous Egypt, a new people, the law of holiness, submission to God, new worship, the procession toward the homeland. The Gospel of John is a reinterpretation of the Exodus: the paschal lamb, the bread from heaven, water flowing from Christ, the healing of those who look upon the crucified one, the "passage" toward the Father, the Pasch. The Book of Revelation underlines the parallel between the sufferings of the Church and the sufferings of Israel; Christians are heading toward a new Jerusalem.1

1.Wilson, Clifford J. "The Exodus in the Christian Bible: the Case for 'figural' Reading" Theologische Realenzykloptidie (TRE) 10.741-45