While I can't speak to the question of what the founders intended, the issue of the relevance of the Bill of Rights in regulating State governments and the use of their "police powers" has been one that has been the subject of substantial Supreme Court jurisprudence. Currently, the majority of the Bill of Rights is applied to the States via the Incorporation Doctrine of the US Supreme Court, but that isn't how things started. (One should note as we go forward that the 9th and 10th amendments are totally irrelevant to this discussion. We only really care about the first eight)

The notion that the Bill of Rights did not apply to the States was enshrined into federal jurisprudence by the decision Barron v. Baltimore from 1833. The lawsuit itself was pretty tame, the owners of a wharf suing the city of Baltimore because the city's actions had diverted certain streams. These diversions, the plaintiffs alleged, filled the nice deep water in front of their wharf with silt and made the surrounding waters shallow. The plaintiffs win against the city at the trial level, but lose on appeal in the Maryland appellate courts. So they appeal their loss at the appellate level to the SCOTUS. They make the case that city's actions constituted a taking under the fifth amendment and they were due just compensation.

Now at this point we should make a brief diversion to explain how appeals from state courts to the SCOTUS works. In general, you can't appeal from a state's highest court to the US Supreme Court just because you lost. Your case has to implicate federal constitutional or statutory law in some way. So, Barron could only have a chance at being successful in his appeal if the Fifth amendment really did apply to the City of Baltimore. If the SCOTUS didn't have jurisdiction, the Maryland appellate decision would stand as the final say in the case.

And the SCOTUS did end up finding it lacked jurisdiction to hear the case because the Fifth amendment didn't apply to the City of Baltimore. Writing for a unanimous court, Chief Justice Marshall says that

"The question thus presented is, we think, of great importance, but not of much difficulty. The constitution was ordained and established by the people of the United States for themselves, for their own government, and not for the government of the individual states. Each state established a constitution for itself, and in that constitution, provided such limitations and restrictions on the powers of its particular government, as its judgment dictated. The people of the United States framed such a government for the United States as they supposed best adapted to their situation and best calculated to promote their interests. The powers they conferred on this government were to be exercised by itself; and the limitations on power, if expressed in general terms, are naturally, and, we think, necessarily, applicable to the government created by the instrument. They are limitations of power granted in the instrument itself; not of distinct governments, framed by different persons and for different purposes."

Now, there's an awful lot to cover when it comes to how we somehow get to applying the bill of rights against the states via the 14th amendment, but to make it brief, let's start with what the Incorporation Doctrine is.

The Incorporation Doctrine basically holds that most aspects of the first 8 amendments apply to the states via the due process clause of the Fourteenth Amendment, which reads that "nor shall any State deprive any person of life, liberty, or property, without due process of law." This doctrine was not created by any single case. Incorporation was a piecemeal process, where individual clauses of the first 8 amendments were applied to the states via this 14th amendment due process reasoning.

The earliest start of this process is arguably "Gitlow v. New York" from 1925, although special mention should go to "Chicago, Burlington & Quincy Railroad Co v. Chicago" from 1897, which held that the taking of private property for public use without compensation violated the 14th amendment due process clause.

Gitlow however is where we really see the Incorporation Doctrine as we'd recognize it today. Arising out of a criminal case, Benjamin Gitlow had been tried and convicted for "advocating the overthrow of organized government" and that he had printed, published and circulated papers advocating the same. The charges stemmed from his involvement with the "Left Wing Section" of the Socialist Party. Although the SCOTUS upheld Gitlow's conviction, it does gives us this rather huge change from the Barron status quo:

"For present purposes, we may and do assume that freedom of speech and of the press which are protected by the First Amendment from abridgment by Congress are among the fundamental personal rights and "liberties" protected by the due process clause of the Fourteenth Amendment from impairment by the States. We do not regard the incidental statement in Prudential Ins. Co. v. Cheek, 259 U. S. 530, 259 U. S. 543, that the Fourteenth Amendment imposes no restrictions on the States concerning freedom of speech, as determinative of this question."

This reading of the 14th amendment due process clause, which saw it as extending the federal freedom of speech to regulate the states, was affirmed in Stromberg v. California. Stromberg struck down a clause of a state criminal statute because it, in its ambiguity, could be construed as prohibiting the symbolic speech of displaying a red flag. As the court noted, "It has been determined that the conception of liberty under the due process clause of the Fourteenth Amendment embraces the right of free speech."

continued in next comment.