Did Christians stop the practice of Berith (Brith, Bris?) immediately and based on which scripture?
This was a topic of dispute in early Christianity, between believers who had come from the Jewish faith, and gentiles. The issue relates to whether Christianity is part of Judaism - which would mean that converts had to be circumcised - or whether it is an independent faith. If it is not needed for adult converts, then it would not be applicable for children either.
Acts 15:1-21 records a council, held in Jerusalem, to decide. This probably took place in 50 or 51AD. Peter said (v7-11, NIV):
Brothers, you know that some time ago God made a choice among you that the Gentiles might hear from my lips the message of the gospel and believe. God, who knows the heart, showed that he accepted them by giving the Holy Spirit to them, just as he did to us. He did not discriminate between us and them, for he purified their hearts by faith. Now then, why do you try to test God by putting on the necks of Gentiles a yoke that neither we nor our ancestors have been able to bear? No! We believe it is through the grace of our Lord Jesus that we are saved, just as they are.
and the council agreed to write to Gentile believers, saying (v28-29):
It seemed good to the Holy Spirit and to us not to burden you with anything beyond the following requirements: You are to abstain from food sacrificed to idols, from blood, from the meat of strangled animals and from sexual immorality. You will do well to avoid these things.
That did not entirely settle the issue, since it says nothing about whether Jewish converts could or should observe the law of Moses, and in addition there continued to be many "Judaizing" Christians that required converts to be circumcised and follow the Torah. Historically speaking, the destruction of the Temple in 70AD, and the Roman slaughter and dispersal of Jews, dramatically reduced their numbers and influence. Though groups such as the Ebionites continued to exist for several centuries after that, "mainstream" Christianity no longer required circumcision at all.
Paul writes about circumcision at great length in his letters, especially in Galatians but also in Romans, 1 Corinthians, and elsewhere. For example, Romans 2:25-29 says:
Circumcision has value if you observe the law, but if you break the law, you have become as though you had not been circumcised. So then, if those who are not circumcised keep the law's requirements, will they not be regarded as though they were circumcised? The one who is not circumcised physically and yet obeys the law will condemn you who, even though you have the written code and circumcision, are a lawbreaker.
A person is not a Jew who is one only outwardly, nor is circumcision merely outward and physical. No, a person is a Jew who is one inwardly; and circumcision is circumcision of the heart, by the Spirit, not by the written code. Such a person's praise is not from other people, but from God.
His theological case is that circumcision is a sign of faith, but it is not the same as actually having faith. For Paul, the outward symbol is unnecessary. More generally, one of his major themes is that adherence to the Law does not guarantee salvation (see Galatians 5, for example).