Are there examples of individuals moving between the Senate and the House in the US government (specifically in the past 20 years)?
Yes, people serve in both the house and the senate. I can't find a list that only lists those that have served in both, but this list of congressmen and women that have served the longest shows plenty of examples of people that have served in both:
Just a few from the the list that are currently serving:
- Ed Markey
- Chuck Grassley
- Thad Cochran
It would be highly unusual for a Senator to become a Representative. Note that there are one hundred Senators and four hundred thirty-five Representatives. So a Senator is about four times as influential in the Senate as a Representative is in the House. Also, loss of seniority would be a problem. The new Representative would start over in the House. The two chambers are themselves comparable in influence, although there are some differences (e.g. tax bills need to originate in the House; only the Senate reviews presidential nominations and treaties).
Loss of seniority is also a problem when switching from the House to the Senate, but the increased influence of a Senator compensates. You ask for recent examples. There were eight new Senators who had been in the House in 2014:
- Tim Scott
- James Lankford
- Tom Cotton
- Cory Gardner
- Bill Cassidy
- Gary Peters
- Steve Daines
- Shelley Moore Capito
There were also six new Senators from other sources: Mike Rounds; Thom Tillis; Ben Sasse; Joni Ernst; Dan Sullivan; and Brian Schatz. That's a total of eleven new Senators in thirty-three regular elections plus the three mid-term replacements. Anyway, over half of the new Senators were from the House. That's consistent with the 2012 result, when seven of fourteen were from the House.
In 2010, there were three new Senators who had previously been in the House but weren't at the time of their Senate election: Dan Coats; Rob Portman; and Pat Toomey. Including those three, that would be eight of eighteen in 2010. That's half in three elections. But in 2008 and 2006, only three of ten were in each (six of twenty total). One of those was 2016 Presidential candidate Bernie Sanders.
It is very common for Representatives to become Senators. It is so common that it would not be work giving examples.
It is uncommon, but not unknown for Senators to become Representatives. Claude Pepper was a Senator from Florida who was defeated and returned to Congress as a Representative. John Quincy Adams served as Senator, then President, then Representative. Henry Clay was a Senator, the speaker of the House, then Senator again. (This is not a complete list. There are some others I know of as well.).
This switch has not occurred recently.