The best replacement for boxing is mma.

The best alternatives to boxing are rated as following:
★★★★★  mma
★★★★  karate
★★★★  muay thai
★★★★  kickboxing
★★★★  ufc
★★★  martial arts
★★  wrestling
★  kung fu
★  wing chun
★  crossfit
★  judo
★  jiu jitsu
★  taekwondo
★  bjj
★  unboxing
  weight lifting
  krav maga
  street fighting
  mixed martial arts
  mma who would win
  aikido
  pro boxing
  muay thai for mma
  boxing
  mma popularity
  tae kwon do
  mma better
  football
  basketball
  ufc ratings
  kickboxing which is better
  ultimate fighting

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Speaking about boxing, one probably shouldn't forget about boxing.

Other search terms related to boxing and their ratings are:
★★★★★ boxing
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Google Trends

Scale is based on the average worldwide traffic of boxing in all years. Click on graph for more.
boxing
mma
karate