would you pick to be transported in time machine, dropped into the modern era (80's to present) as is, and do the best?
The further back you go, the less the fighters become a known quantity. For example Sullivan or Jeffries might have beaten every champion after them, or they might have lost to men like Mike Weaver. I would say that Dempsey is a very known quantity, and I am comfortable saying that he would have had success in later eras. With Johnson it becomes more speculative, and anybody before Johnson, if becomes very speculative.
Dempsey is the obvious one, but i go for Johnson too as we would see Johnson with focus on his boxing/fighting rather than having the race issue involved too
Jeffries easily. He had the size, power, and durability. The technique of his times was what they were, but with a good 1980's trainer, that would not be an issue as Jeffries was a well-coordinated and quick athlete.
Bob Fitzsimmons, since the OP never said at Heavyweight. At Heavyweight I'd pick Johnson, then Jeffries. Dempsey has the better style for the modern division, but I think those two were just better.
Dempsey style would not change. His defense on film is mediocre, and his lack of competition doesn't help. He could have fought Wills, so we could see how he handles a puncher around 215 pounds, and Greb to see how he handles a much quicker fighter. Neither happened. Tunney ( the best he faced ) outboxed Jack easily 19 of 20 rounds, and I don't think Jack was too far gone to suggest Tunney wasn't going to beat him if he was at his best. Brennan, who wasn't that good gave Dempsey a really hard fight, Jack had to come from behind to win. George Carpentier who was small was holding his own and even landed a stinger, then hurt his arm/hand ( Forget which one ) ad quickly faded. Firpo who looks awful on film managed to floor Dempsey twice, and based on rules could have been the TKO winner as Dempsey was helped back into the ring. The editing in that film smells. And Jack Starkey gave Dempsey a pasting until Dempsey foul then took advance of Sharkey complaining to the ref to delver the KO blow. And then there is the alleged glove incident vs Willard. Nothing was clean here, too much smoke and too many uneven moments for Dempsey. Pre title he was KO'd in 1 vs Flynn, and had his hands full at time in other matches. What Dempsey had was a heck of a promoter and an exiting style when the filming process really began to improve.
Dempsey, in modern day imo would do well. He was ahead of the game back then, put him in with today's top trainers, he'd be knocking guys out left, right and center!
He was a drunk. But he could whack. And that's translatable. The obvious answer is Dempsey. I just wanted to give an alternative.