Computer magazine MacAddict, one of the two magazines split off from the defunct CD-ROM Today in 1996 (boot, now Maximum PC, was the other). When it started out, MacAddict was unafraid to have fun: they often included little cartoons in the letters section and back page (even a stick-figure mascot, Max, who was also used in their reviewing scale); the pages were bright, colorful and rife with Running Gags (for several issues, they joked that each magazine was soaked in Downy before it was shipped out); the CD that shipped with every issue would include something funny like a video of the staff destroying a Windows computer; and so on. In the early 2000s, the magazine got a white, sterile makeover (replacing the Max scale with a normal five-star scale), and the tone gradually shifted to a far more serious and straight-laced approach. This shift culminated in 2007, when the magazine was renamed Mac|Life.
Older than Television: In 1929, Wash Tubbs went from "bigfoot" humor to high adventure with the addition of soldier-of-fortune Captain Easy to the cast.
Another early example is Skippy, a comic strip from the 1920's through 1940's. It was originally a wildly popular comic about a mischievous kid, but it started getting more and more serious and political when creator Percy L. Crosby became convinced that President Franklin Roosevelt was a Communist. Eventually, a company with connections to the IRS used several "random" audits to successfully take over the rights to the name Skippy. The company was, of course, the maker of Skippy peanut butter. Crosby ended up suicidally depressed in a mental hospital. You can read the whole story here.
The strip which eventually became Steve Roper & Mike Nomad began life in 1936 as a wacky comedy starring a stereotyped American Indian named Big Chief Wahoo. Roper was introduced in 1940 and took over the strip, until by 1947 Big Chief Wahoo had been written out and the wacky humour entirely dropped in favour of action adventure. Mike Nomad appeared in 1956, by which time the original nature of the strip had totally vanished.
Ironically, Big Chief Wahoo had not been planned to be the strip leader; he was supposed to be a supporting character to the Great Gusto, a traveling salesman/conman. Wahoo was instantly much more popular and Gusto, reduced to second banana status from the beginning, was gone by 1939.
Funky Winkerbean literally jumped (in the form of a 10-year timeskip) from a high-school based gag strip (with occasional dramatic Very Special Episodes) to a frequently depressing drama strip where Anyone Can Die. A second ten-year timeskip seems to have abandoned all pretense of zany (or should that be funky?) comedy, preferring a more down-to-earth kind (when, that is, there's any at all).
Also, ghost voyeurs.
Luann used to be a light-hearted, Gag-A-Day look at the ups and downs of a 13-year-old girl dealing with friends and family. Now...
9 Chickweed Lane started life in 1993 as a gag-a-day strip about 3 generations of females and their daily experiences. It has since become a piercing look at personal relationships and the human condition, with its recent "mega-arc" - encompassing the lives of many people - lasting several years.
Doonesbury always had a political element, but in its first couple of years in national syndication it was mostly a light-hearted strip about college life (continuing where Garry Trudeau's work at Yale left off). Once Watergate happened it focused more and more on politics. On top of that it became more of a serial strip, and even introduced Anyone Can Die to the comics page.
Bloom County started out as a rural humor strip, but as time went on they started adding more and more political and pop culture satire, which would dominate the strip for the rest of the run. Strangely, its Cerebus syndrome coincided with it sliding down the Sliding Scale of Fourth Wall Hardness all the way to having No Fourth Wall.