I do have to ask: What conditions are you referring to? What has actually changed that would give rise to a different understanding of the meaning of "due process"?
As I said, I am not suggesting that the literal meaning of the term "due process of law" has changed. What I am suggesting is that conditions and standards have changed such that we must conduct ourselves differently than we did in the 1860's if we wish to avoid depriving individuals of life, liberty or property without "due process of law." Just as our conception of what constitutes a "suitable education" or "appropriate healh care" has changed since the 1860's, so has our conception of what constitutes "due process of law."
Take for example the right to counsel in criminal cases. In the 1860's, it was not felt that a criminal defendant's right to counsel was an essential element to what that generation viewed as a fundamentally fair criminal justice system. However, on behalf of those folks, it should be understood that many of the procedures and most of the evidence in those days (typically the testimony of lay witnesses) was not nearly as complex as they often are today. They just didn't have scientific evidence such fingerprints, DNA, recordings, etc. So, for one thing, the entire system has simply become vastly more complex.
However, beyond this increasing complexity, I think an additional factor concerns an evolutionary change in our basic conception of what should be expected of a state which desires to deprive an individual of his life or freedom. You can see the Supreme Court's opinion in Powell v. State of Alabama (1932) 287 U.S. 45 to get an idea of what a majority of the Alabama Supreme Court then thought to be a constitutionally adequate system for depriving persons of their lives. I seriously doubt that any current member of the Alabama Supreme Court would consider the system described in Powell as an fair or adequate method for a state to determine who shall live and who shall die. More than thirty years later, the Supreme Court in Gideon v. Wainwright (1963) 372 U.S. 335, held that the right to counsel is essential to a fair trial in every significant criminal case: "The right of one charged with crime to counsel may not be deemed fundamental and essential to fair trials in some countries, but it is in ours."
One aspect of all this doesn't change. Thirty years from now, another generation of Americans will have to determine precisely what "due process of law" requires in their time.