This is fairly difficult to gauge in U.S. politics because of the ongoing process of realignment.
In 1950, Southern whites were predominantly Democrats, while blacks in the South were predominantly Republicans. These days, of course, those positions are reversed. Southern whites were alienated from the Democratic party when it embraced the Civil Rights movement and gravitated towards the only viable alternative, the Republican party, instead, first in federal elections, and eventually in state and local elections.
Likewise, as recently as the 1970s, the Republican party was thriving in the Northeast, while today, only a handful of Representatives and Senators in the U.S. Congress from the Northeast are Republicans.
Many people attribute this shift to Republican Presidential nominee Goldwater's "Southern Strategy". Candidates like Eisenhower, Nixon and even Ronald Reagan (who was an actor and a union President) would all probably be too far to the left to have any reasonable shot as winning a Republican Presidential nomination today.
As recently as the 1980s, at the federal level, the U.S. had what amounts to a three party system with Southern Democrats swinging between Republicans and Northern Democrats - favoring Republicans on military and social issues, and favoring Democrats on economic issues. Today, there are almost no moderates in Congress with all Democrats being more liberal than all Republicans on almost every issue.
People's political views have been considerably more static than their party affiliations.
UPDATE ADDED August 19, 2020:
Another confounding factor is that in absolute terms, the political center has consistently moved towards the left over time.
For example, in the 1950s segregationist members of the U.S. Congress, who were in tune with their constituencies, railed against interracial marriage, and in the 1960s civil rights icon Martin Luther King, Jr. while outspoken on many civil rights issues was loath to focus on the "social question" of interracial marriage. But after the U.S. Supreme Court decided the case of Loving v. Virginia declaring laws that prohibited interracial marriage to be unconstitutional, public opinion changed. And, by the 1980s, formerly segregationist U.S. Senators who had decried interracial marriages as abominations had Congressional staff members who were in interracial marriages.
In general, the political leanings of particular geographic regions tend to be very stable over time frames of hundreds of years on whether they lean conservative or liberal relative to a nation as a whole on all manner of diverse issues. But a position that would be considered to be a decidedly liberal position at one point, such as a policy position that same sex couple relationships should not be legally recognized but should also not be criminalized, may at another point in time become a decidedly conservative position.
While there have been ebbs and flows back and forth in more conservative and more liberal directions, in the long run, on the scale of decades or longer, there has been a consistent secular trend towards more liberal political views.
So, if you are measuring a particular individual's liberal v. conservative leanings on a basis relative to others, rather than in absolute terms, older people tend to become more conservative over time simply because their views, more or less solidified when they became politically aware and first seriously considered these issues may not have changed as rapidly as the views of the general public as a whole.
Since political parties must constantly renegotiate the terms of their stance on the liberal to conservative spectrum in absolute terms, as part of their party platforms, in order to continue to have a viable chance of securing majority legislative support for some or all of their agenda, the partisan dividing line between political parties, all other things being equal, tends to encourage someone whose political views in absolute terms remain constant or changes less quickly than the population as a whole from left leaning parties to parties to the right of their original political party affiliations.
Examples other than civil rights of issues which have shown secular trends to the left over time in the modern U.S. include alcohol prohibition, marijuana legalization, support for mandatory minimum sentences for crack cocaine offenses, support for environmental regulation, support for stricter regulation of police misconduct, support for legalization of hormonal birth control, tolerance for out of wedlock births, opposition to corporal punishment for crimes and misconduct, reductions in the stigma associated with divorce, acceptance of profanity, violence and nudity in media oriented towards children, recognition of the health risks of smoking, support for workplace and public safety laws, acceptance of government subsidized health care programs, recognition of martial rape as a crime, and tolerance for non-Christian religions.
But because of the realignment process in the United States (e.g. in 1919-1920 the 19th Amendment to the U.S. Constitution giving women the right to vote was opposed most strongly by conservative Democrats in the South and had much wider support from mostly Northern and Western Republicans), however, the natural tendency of people's views on particular concrete policy positions to be viewed as more conservative over time, this subtle and gradual effect is swamped by realignment and impossible to disentangle by looking at changes in political party affiliations and identities in the 20th century.
