CMLL will edit any blood it can out of any of its televised matches or simply not air them if it cannot.
Around the turn of the century, there was a memo circulated by Turner Broadcasting for CNN that declared that "foreign" stories be referred to as "international" stories. WCW received this memo, which wasn't meant for them. This lead to the term "foreign object" (a weapon that shouldn't be part of the match, e.g. a chair at ringside) being changed to "international object." Back in WWF/E Jim Ross called weapons used in matches "international objects" on at least one broadcast as an in-joke.
Also in WCW, when the Insane Clown Posse were working there, the "Violent" in Violent J's name was deemed unacceptable, causing the commentators to refer to him as Jake Jeckel. Sometimes. Other times they called him Violent J.
When 'asshole' chants first became a thing in professional wrestling, Turner Broadcasting tried very hard at first to figure out a way of censoring the chants. Since there was no way to get the fans to stop and trying to mute the chants without muting whoever was talking was impractical, they gave up after a couple months.
During the early 2000s, UPN forced WWF to rename two of its wrestlers.
"Bad Ass" Billy Gunn's name was deemed problematic because of the gun reference. He went by Mr. Ass for a few years. (Yes, "gun" was deemed more offensive than "ass." Weird.)
Val Venis' name was deemed unacceptable because it rhymed with penis. One of his nicknames had been 'The Big Valboski', and that briefly became his real name.
WWE's Divas Undressed special. Even though it aired during the graveyard block, they still spent a lot of time cutting away to judge reactions during particularly racy bits.
In one particularly disappointing example, the heel character of Muhammad Hassan was completely written out of storylines following the London terrorist bombing of the summer of 2005. This despite the character referring to himself as an Arab rather than a Muslim (although his ring entrance did make liberal use of Islamic imagery) and using a garrote made of piano wire as a heel weapon (a more stereotypically Italian way of murdering someone, and in fact Hassan's portrayer, Mark Copani, was partly Italian-American).
Indy wrestler Dalton Castle's gimmick includes his being accompanied at all times by two men who form human furniture for him and fan him off during matches. When he made a few appearances in TNA in 2015, they had him accompanied by two women instead, although otherwise they left his gimmick intact.