Seems to me Arena Football, Indoor soccer, and so many other “pro league” schemes seem to be so “fly by night”. Are there any examples of American/Canadian “pro” franchises that have continually survived year after year? Minor legue teams financed by major league teams don’t count!
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What do you mean by “‘pro-league’” in quotation marks?
By definition, a “farm team” is supported by a major league team.
Arguably the Atlanta Braves can be traded back to the 1869 Cincinnati Red Stockings.
The Chicago Cubs can be traced back to the 1871 Chicago White Stockings
The Montreal Canadiens were founded in 1909
The Sacramento Kings were founded as the Rochester Seagrams in 1923
The Green Bay Packers and the Decatur Staleys (now the Chicago Bears) were founded in 1919.
Three NBA teams - New York Knicks, Boston Celtics, and Golden State Warriors - have been around since professional basketball started in 1947. The Warriors originally played in Philadelphia.
I think the OP is looking for an independent professional team, not in the major leagues. I’d argue the Harlem Globetrotters, founded in the 1920’s, would be a possibility depending on your definition. You could also consider The King and his Court, a 4 man softball team anchored by King Eddie Feigner who took on all comers from 1946 to the early 2000’s.
There are also many Independent Baseball leagues but they come and go rather frequently.
The NBA started then but professional basketball had been around since the 1920s.
It seems to me that the handful of pre-major league clubs that have survived into the modern day are the ones that joined the major leagues, or helped to found them.
https://www.basketball-reference.com/ only goes back to '47. 
The American Basketball League, formed in 1925.
Okay. I’m guessing your emoji indicates you realize that this doesn’t contradict what I said?
Plus, professional sports generally predated modern-style organized leagues.
Not questioning, just pointing out that there seems to be no easy access to stats from then.
Both the Packers and the Bears, which were originally independent teams, predate the founding of what became the NFL (founded as the American Professional Football Association in 1920) – the Staleys / Bears were a founding franchise in the APFA, and the Packers joined in 1921.
However, the oldest NFL team is actually the Arizona Cardinals. They were founded as the Morgan Athletic Club, on the south side of Chicago, in 1898.
And then we get into issues of what constitutes “the same team”. The Arizona Cardinals have neither the same name nor the same location as the Morgan Athletic Club, and they certainly have no personnel in common: It’s a Ship of Theseus.
Evidently, the Cardinals that had been the Athletic Club didn’t even exist between 1906 & 1913.
The CFL Toronto Argonauts claim, not unquestionably, to be the oldest individual franchise still extant, having dated their beginning to a recreational exercise of the Argonaut Rowing Club in 1873.
I did not know this until now, but it does appears to be correct. It does look like the Cardinals can, at least, claim to have bee continually in existence from 1913 on.
The Toronto Argonauts were not pro until much later, correct?
OTOH the Chicago White Stockings, founded in 1870, are the same organization that is now known as the Chicago Cubs and has always been professional.
The White Stockings didn’t become the White Sox?
No, the Chicago White Stockings became the Colts, Orphans, and then Cubs. In 1900, the Saint Paul Saints of the Western League moved to Chicago and swiped the White Stockings name, which had been largely abandoned by the Cubs, but still held sway. In 1901 the Western League became the American League.
The same thing happened in Boston. The Boston Red Stockings were a successful club in the National Association, and then in the National League (they were founded by the same manager who had founded the Cincinnati Red Stockings in 1869).
The Boston N.L. club used many different nicknames over the years, but by 1908, they decided to give up their red uniform trim for blue. At that point, the American League’s Boston club (established in 1901) took the opportunity to claim the Red Sox name. The National League team eventually settled on the name Boston Braves, and took that name to Milwaukee and then Atlanta.
The Baltimore Orioles name had also been the name of a National Association and National League club before an American League team picked it up.
Who, I believe, brought along approximately half of the Cincinnati club with him.