It is probably the oldest tradition of baptism in the Christian church. Early writings (notably the Didache) specify that baptism should occur in living/running water such as streams and rivers. Biblical sources (e.g Matthew 3:6, Acts 8:38) are also sometimes read as describing full immersion in rivers and other bodies of water.

Immersive fonts and dedicated baptistries are rare before the Edict of Thessalonica and widespread Roman adoption of Christianity. Once large, ornate, permanent structures become the norm for churches, baptism becomes more common in or near the church structure itself. There are early examples from Dura Europis and the San Ponziano Catacomb (both 3rd cCE), and they become more common especially in the East. As Christianity becomes a major religion with it's own symbolic architecture, it moves away from using available places in nature and begins constructing ritual spaces to replace them within churches. Source.

I know damn near nothing about Christianity after the fall of the Western Empire, so I can't help you much with the modern American tradition. Baptismal tradition definitely varies by sect, and I know full immersion is a thing for Baptist/Anabaptist denominations in the US. I don't have the background knowledge to trace out the full list of schisms, sectarianism, and theological debate behind all those different traditions though.