Studies have shown that rank-and-file Democrats have moved leftward much more than rank-and-file Republicans have moved rightward[0], but at least one study shows that Republican representatives have tacked rightward much more dramatically[1].
I've seen [0] before and you can sort of explain [0] and [1] by adjusting the frame of reference. As [0] shows, Republicans have pretty much been unmoved for 30 years. At first view you might conclude it's democrats who are moving right, but if you consider the rest of the 1st world as some kind of "average", I think you would find Republicans are moving farther right. Issues like health care, social safety nets, and worker protections are pretty much close to "center" in Europe, but are "left" in the US.
One could argue Republicans moved much father rightward of the 1st world mean than democrats have, which reconciles with [1] - in order to stay mostly politically unchanged after 30 years you must be more and more steadfast.
Democrats have always outnumbered Republicans, but the gap was much wider mid-century. The gap has narrowed considerably, but mostly because of declining Democratic party affiliation. Independents now outnumber Democrats.
[0]: https://www.pewresearch.org/politics/interactives/political-...
[1]: https://www.washingtonpost.com/news/wonk/wp/2015/06/02/this-...