It has been common to move PCs since the game's early days
While one example is related in the Oerth Journal as outlined in this answer (in this case Gary Gygax and Rob Kuntz bringing their Greyhawk characters to play in the City of the Gods dungeon/adventure created and run by Dave Arneson), the practice of bringing one's characters into a different DM's dungeon from another DM's dungeon was referred to in both Strategic Review and in the Dragon magazine. (And anecdotally among the games early players).
Personal experience: we did it with great frequency
I saw it a lot during play in our groups over the first few years I played (1975-1979) in high school and college. But there were some issues that inevitably arose. As I allude to in this answer, a DM's toughness would vary as would the propensity for distributing magical treasures. This could lead to a PC showing up with very few magic items in a game where all of the other players were loaded, or the reverse: a PC loaded with magic items in a game where they were scarce. There was no standardization, there was no WBL (a feature of 3rd edition). A few points from that answer before I proceed:
Gygax said this in his explanation of Vancian Magic (Strategic Review,
Volume 2 Number 2 (7th and last SR). He addresses it also in his
article "D&D is only as good as the DM." (p. 22 of same issue)
... players ... in the 20’s, 30’s and 40’s of levels ... have not really earned their standings, and their actual ability has no reflection on their campaign level, they are easily deflated (killed) in a game which demands competence in proportionate measure to players’ levels. {citations in my linked answer}
The disparity in DMing as a tempering tool for PCs led to charges of "{this DM} is too easy" and the epithet "Monty Hall" being applied {named for a game show host)
This volume is something else, also: our last attempt to reach the
"Monty Hall" DM's. Perhaps now some of the 'giveaway' campaigns will
look as foolish as they truly are. This is our last attempt to
delineate the absurdity of 40+ level characters. When Odin, the
All-Father has only(?) 300 hit points, who can take a 44th level Lord
seriously? (Eldritch Wizardry, Foreward)
These were potential obstacles to portability, but these weren't necessarily obstacles: that depended on negotiations between the DM and the Player.
Potential problems for a receiving DM
The DM, we learned, had to vet a PC that a player wanted to bring into to their game, particularly for magic items and custom magic items. The concept of "balance" was notional and subjective, but if a 4th level Fighting Man had +2 Armor, a +2 Shield and a +2 Sword that had extra damage against undead, and a ring of regeneration wanted to come into the game in a group from levels 3 to 6 where none of the other PCs had that level of magic item accumulation, the other players could resent the port. (Or they might not care). Likewise "I want to bring my 8th level Magic User for your next raid" to a of 3rd and 4th level PCs might create a power delta, but the group might be super receptive to having the new ally. We had groups with disparate levels, but overpowering the group would still manifest with varying reactions by the other players.
As we DM'd we learned how to vet moving PCs: for example we might deactivate a favorite magic item (such as the 6th level player with an artifact) for our sessions, or, we might ask the player swap it for something we felt fit our world better.
With the above considered, sometimes the DM would port the PC in and let the chips fall where they may.
Potential problems for players in that original scheme
The prime pothole in the road was a lack of familiarity with 'how we do it at this table' since every instance of D&D was very much customized by the DM. You'd learn as you go. It was not uncommon to see annoyance among some other players if the above vetting had not taken place and the new player didn't fit with that table's style very well, but usually it was a case of the new PC being welcomed.
Potential problems for a game club
I saw a number of unfortunate cases of personality conflicts between players resulting in the banning of one club members' PCs in the DM's dungeon, and another no-kidding 'banishment' of a player from all of the games of all DMs (nearly everyone DM'd a bit) in that club. (Sad but true). He ran a few solo adventures for me now and again (I recall one that lasted most of a weekend when the bulk of the game club were at a Con. I didn't have the baggage the others did, they'd known him longer).
What Adventurers League does is a nice feature of the current edition
Up to Season 7 (where my experience with AL ends) the Adventurers League guidelines (available from WoTC and at DMs Guild) for each season provides a workable means for moving AL PCs from one table to another. While it has its own bugs and potholes (and I gave up on AL when Season 8 began, why is off topic for this answer) it makes the ability to move one PC to multiple tables a reality. I tip my cap to those who DM for, and who host, AL events; it makes PC portability work reasonably well.
With all of the above considered: DM's call
Some DMs just prefer that a character be created to fit their world. In most of the AD&D games I played and DM'd we did that; the fluid 'table to table' movement I experienced was mostly in OD&D play. A good reason to do that is to make sure that the PC fits into the DM's game world as well as the existing PCs. Nowadays, with chargen being so easy/concise, it appears to be a best practice, and is (experientially) how all of the games I have been in have worked out: make a PC for this world/table.