What process did the supreme court settle on to incorporate provisions of the bill of rights as binding on the states?

The answer is selective incorporation

This is a constitutional principle that ensures that states cannot pass legislation that would deny people their constitutional rights as enshrined in the Bill of Rights. This is accomplished through the capacity of the federal government to limit state powers

The supreme court used the doctrine of selective incorporation

The doctrine was one among others that ensured the supremacy of the USA Constitution. The doctrine stated that no state in the union could pass laws that contravenes the bill of rights. The doctrine limited the ability of states to make laws.