How does Barron v. Baltimore tie into Mapp v. Ohio and Weeks v. US?

The three cases show the evolution of the Supreme Court's understanding of how the Bill of Rights should be applied. The Bill of Rights was drawn up to protect the rights of states and individuals from the encroachments of an over-mighty federal government. The landmark case of Barron v Baltimore (1853) established the important legal precedent that the Bill of Rights did not apply to actions carried out by state governments.
In Weeks v United States (1914), the Supreme Court ruled that a seizure of items from a private residence without a warrant constituted a violation of the Fourth Amendment. But, as the case involved the federal government as defendants, the original scope of the Bill of Rights remained. Nevertheless, Weeks was a significant milestone in civil liberties jurisprudence, as it established the so-called "exclusionary principle." This means that incriminating evidence can be excluded from trial if it was obtained via unconstitutional means. The Court's introduction of the exclusionary principle, though related to the actions of the federal authorities, opened the door for a similar challenge to be made in future against state authorities whose actions violated the relevant provisions of the Bill of Rights.
It took quite some time for this to happen, but the Supreme Court duly delivered in Mapp v Ohio (1961). The facts of the case were similar to those in Weeks: incriminating evidence had been seized by the relevant authorities in a manner that violated the plaintiff's constitutional rights under the Fourth Amendment. In its ruling, the Court extended the exclusionary principle to state authorities, whose actions had not previously been covered by the Bill of Rights. In doing so, the Court incorporated the relevant provisions of the Fourth Amendment—which only applied to the actions of the federal government, prior to this case—into the due process clause of the Fourteenth Amendment, which applies to the actions of individual states.