The most well known examples of boxers moving into MMA are the silly one-and-done careers of James Toney, who was demolished by former UFC champion Randy Couture, and Ray Mercer, who punched out former UFC champion Tim Sylvia.
In reality, the "boxing vs. martial arts" argument has a long, storied and complex history. Frankly, it is too long to go into comprehensively, so for the sake of brevity, we will only go over the few examples of success and failure that this writer deems particularly relevant.
It likely comes as no surprise that the first case to look at here is Gene LeBell vs. Milo Savage. For those who are unfamiliar with the fight, in the 1960s a men's interest magazine named Rogue ran an article entitled "The Judo Bums," which claimed judo was not a legitimate form of hand-to-hand combat and offered a $1,000 purse to any judoka who could defeat a boxer in a fight. LeBell took them up on it and was matched up against an aged star in Savage.
The fight was something of a circus act, complete with allegations of greasing and brass knuckles. The actual contest would ultimately stretch across five rounds and be mired by reluctant engagement. The bout lasted until LeBell scored a takedown, took Savage's back and choked him unconscious—Sports Illustrated's Josh Gross told the whole story here.
At the time, though, it wasn't possible for a boxer to find a top-flight camp of judoka to prepare with. The same goes for the Muhammad Ali vs. Antonio Inoki match of the 1970s. In today's combat sports world, an elite boxer would have no trouble finding somewhere to hone other aspects of his game for a potential run in MMA.
For an example of a high-level boxer who actually made a relatively serious run, look no further than the career of former Ronda Rousey protege, Jessica Rakoczy.
While Rakoczy is not a boxer on Mayweather's level—who is?—she worked her way to the No. 5 spot at 125 pounds by amassing a 33-3 (1) record and a small pile of belts. When she decided to take the MMA plunge, her pedigree got her matched up with some of the top female fighters in the strawweight and flyweight divisions. The results were not good.
Her first three opponents were Michelle Ould, Zoila Frausto and Felice Herrig, who are all regarded as some of the best fighters in either the 115-pound or 125-pound weight classes in women's MMA. Ould and Frausto, both formidable grapplers, took Rakoczy down and finished her. Against Herrig, a muay thai kickboxer, Rakoczy was more competitive but came out on the wrong end of a split decision.
She would face weaker competition from there and, combining her amateur and professional fights, has gone 5-1—one of those wins was overturned to a no-contest due to a drug test failure. The early ugliness in her MMA career and recent success show that a strong boxer can be successful with legitimate, dedicated training.
What, though, should fans make of the male vs. female dynamic in a Rousey vs. Mayweather bout?
While YouTube is full of intergender "fights" of questionable legitimacy, there have been several sanctioned boxing and kickboxing matches between a man and a woman. Traditionally, the bouts involve relatively seasoned women beating up inexperienced men with little difficulty (you can check out some of the examples here and here) that, frankly, can be dismissed out of hand. While a promoter tried to pull off a serious woman vs. man match between Ann Wolfe and Bo Skipper in 2005, the bout was delayed due to Hurricane Katrina and was never rescheduled.
Worth watching, though, is how Rousey dominated TUF 17 sensation Uriah Hall in a grappling session in 2013. Hall, who comes from a kickboxing background, fights at 185 pounds and likely walks around well over 200 pounds between fights. That didn't stop her from utterly imposing her will on him when they rolled and repeatedly sinking in her signature armbar.