After Scalia's death I spent a fair amount of time researching this and wrote a post that tried to outline the arguments as best as I could find them. You can see that post here, titled "I Think Scalia's Originalism Is Like the Intelligent Design of Constitutional Theories."
In fact, I recommend that you do look at the previous post and consider this attempt at a follow up to it, as more of a timely reintroduction to the topic and an easy way to get to the original post.
Basically, originalists argue that they are sticking to what the writers of the constitution actually wrote. But then some, like Scalia, acknowledge you can't do it completely literally, it needs some interpretation. Originalists themselves debate over how to interpret the constitution. From James E. Fleming's Living Originalism and Living Constitutionalism as Moral Readings of the American Constitution (pp.1174-1175):
There are numerous varieties of originalism, and the only thing they agree upon is their rejection of moral readings. Some of the varieties include the following. It all began with conventional “intention of the Framers” originalism.17 Then it became “intention of the ratifiers” originalism.18 Of course, we also have “original expectations and applications” originalism (what I elsewhere have called “narrow” or “concrete” originalism).19 Then came “original meaning” originalism, which was refined as “original public meaning” originalism (officially, this is now the position of Scalia and Barnett).20 Scalia himself distinguished “strong medicine” or “bitter pill” originalism from “fainthearted” originalism.21 Then came “broad” originalism (advocated by Lawrence Lessig and many others).22 Now comes “the new originalism” (so characterized by Whittington) as distinguished from “the old originalism.”23 Finally, we add “abstract” originalism (which some, including Whittington, have attributed to Dworkin).24 And we must not forget Balkin’s “method of text and principle,” a form of abstract originalism.25 Indeed, Mitchell Berman has distinguished seventy-two varieties of originalism in his tour de force, Originalism is Bunk.26
Originalists argue that originalism is more pure than the living constitution approach because it is based on the words of the constitution or the intent of the drafters, etc.. The critics of originalism challenge that assertion. The words, they say, are sometimes specific and sometimes general and those words are not clear when it comes to applying them to twentieth (and twenty-first) century dilemmas. They beg for more guidance.
The originalists argue that the living constitutionalists simply insert their own values to interpret the Constitution. Living constitutionalists dispute that, saying they use the tradition of case law which considers past decisions. There is a lively debate in the literature on the basis, say, for overturning old decisions.
The living constitutionalists argue that even the first justices didn't simply read the text of the constitution. Stephen M. Feldman writes in this 2014 BYU Journal of Public Law article :
"Early judicial opinions and legal treatises reveal an eclectic or pluralist approach to constitutional interpretation; no single interpretive method dominated. Early judges and scholars invoked not only reason, but also the text, constitutional structure, framers’ intentions, original public meaning, and so on. Yet, no judge or scholar maintained that constitutional meaning should be ascertained pursuant to a reasonable-man standard."
The living constitutionalists even cite comments from Thomas Jefferson, in a letter, that argues that every generation should write its own constitution, that one generation cannot lock all future generations to their constitution. They call this the Jefferson problem when they challenge the originalists. I've got more on this in the original post.
One of the arguments living constitutionalists make is that the constitution includes both very specific language and abstract, more aspirational language. I'm taking this a little out of context, but this quote gets to this point:
First, Balkin’s method of text and principle conceives the Constitution as embodying not only rules but also general standards and abstract principles.32 He, like Dworkin and I, rejects efforts by originalists to recast abstract principles as if they were rules (or terms of art) by interpreting them as being exhausted by their original expected applications.33 In interpreting these general standards and abstract principles, we have to make moral and political judgments concerning the best understanding of our commitments; history alone does not make these judgments for us in rule-like fashion.
That comes from Fleming again (p. 1176).
Here's a book review of David Strauss' The Living Constitution. Strauss is one of the authors that Fleming is addressing in his article. The review tries to outline the arguments in somewhat simpler terms.
As you can tell, I'm a little overwhelmed by trying to sum this up. I think the previous post on originalism does a better job of giving a holistic view of originalism. Living constitutionalism is touched on, necessarily, in that post. I was hoping to give a better look at and critique of the living constitution model here, but that just isn't going to happen. Too much going on to get it done. And given that Gorsuch is in the spotlight now, this seems a good time to post this and point you to the original post.
This all fascinates me as it fits neatly into this blog's underlying theme - how do you know what you know? How do supreme court justices know what the constitution means and how to apply it to specific cases. That's precisely what this debate between originalism and living constitutionalism is all about.