Chess games are largely broken down into three phases: Opening, Middlegame, and Endgame. The opening is largely dictated by theory--through years of analysis (and hundreds of years of material), strong players know how to make many moves before they really have to start analyzing. We call this "preparation". A strong grandmaster could make their first 20 moves based on analysis and previous study. If you want to see this in action, watch GM Peter Svidler, one of the strongest players in the world, analyze a game played between Levon Aronian and Vladimir Kramnik about 10 days ago (it was probably the best game of chess played so far this year).

We have the benefit of years of analysis, work with chess engines, etc. to hone all this opening theory. Bobby Fischer, who was the best player in the world in 1970, would be hard-pressed to come out of an opening with a decent position against a modern GM. A chess grandmaster from 100 years ago would have an even tougher time. Frankly, most chess masters from previous eras, even world champions, would be crushed in the opening by modern grandmasters.

But if they could get to a middlegame, they'd have a decent shot. No one can question the chess genius of Paul Morphy. Though he was active in the 1850s and 1860s, Morphy could easily beat many modern grandmasters, if he reached the middlegame on equal footing. The same is true for Steinitz, Lasker, Capablanca, Alekhine, etc. But those players were, in their prime, leaps and bounds better than their contemporaries.

If they did not decide the game in the middlegame, they would likely lose an equal endgame. Endgames are algorithmic, and lend themselves to rote memorization of patterns, key squares, etc. We have many books on endgames (e.g. Dvoretsky's Endgame Manual) which simply did not exist fifty years ago.

Now if you took one of the best historical players, and gave them access to modern knowledge, they'd certainly beat some modern grandmasters. I have no doubt that Bobby Fischer's work ethic rivaled or surpassed any world champion from the past five decades. Performance in chess, like most anything else, is a function of access to training and time. Our predecessors lacked the former, and considering the paucity of professional chess players pre-20th century, one could argue they lacked the latter as well.

If you're interested in learning more about the players I cite, I've done biographical sketches of most of them. Just search YouTube for "Life and Chess of" whomever you're interested in.

Oh a question I can answer! In this case it's notable that modern chess experts would be at a huge handicap in the medieval era, because the game wasn't the same. The Vizir (Queen in European version) could move 1 space in any direction, and the Fil (Bishop) could move up to 2 spaces diagonally, but could jump over other pieces. In addition, Pawns couldn't move 2 squares on their first move, and there was no "castling".

This makes the game slower, and far more positioning based, since only the Rooks can move more than 2 spaces at a time. Pieces sweeping across the board early is simply impossible since at minimum it takes 4 moves to free a rook from the back rank (double advance a pawn, move the rook forward, move the rook over). In addition that is most likely far from the most important series of moves, since the side pawns are traditionally considered less valuable to advance than the middle ones (as they cover less ground and support less strategies. Most likely with no castling and no double advance on the first move, rooks would be stuck in the back line until midgame.

Now it's a simple fact that the strategy of chess is largely based on memorized play patterns - all of which goes flying out the window when the game works completely differently. Now I doubt the original game had some of the systemic study that we've poured into modern chess, but at the same time the systemic study we've applied is completely useless when faced with a vastly different rule set. It's not apples and oranges, but it is two distinctly different types of apples.

I really wouldn't underestimate the Indian players who created the game, and think that the modern chess grandmaster would be at an extreme disadvantage even if given several months to learn the variations between ancient and modern chess. Even small rules changes can have large impacts on memorized openings, and the scope is anything but small.

Finally a question that I can answer with some authority. The first manual of chess that we're aware of is by a Spanish priest in 1561. Ruy Lopez the author is also credited honored by having a popular opening named after him.

Since that time we have a history of chess and games played but it really didn't have a lot of advances in theory until the 1800s. And there we find a man named Paul Morphy who may well be the first chess Genius. Paul's games are certainly still studied today, as they present not only examples of how to gain and keep initiative and they represent an older style of playing where sacrifices are accepted and we get to see examples of material or piece count and it's relative value versus the activity of the pieces.

Grandmaster Ben Finegold has given some excellent lectures on Paul Morphy on youtube and his performance in blindfold simultanous exhibitions shows a talent that many masters of the game, don't currently display. Master in this case I will note as being rated over 2200 according to FIDE which is the world body for chess.

So we'll have names like Alexander Alehkine or Jose Raul Capablanca who's games are still studied but they're relatively modern living in the early 1900s.

Two books that I can recommend if you want to know how we stack up current vs previous world champions are Timman's Titans by Jan Timman or My Great Predecessors by Garry Kasparov. The latter book was written by a world champion so he'd know who stacks up

As has been noted by /u/thepromisedgland there are newer methods known to us now that weren't available previously. Modern computers are treated as if they'll give the "Objective" best move in any situation and when you can memorize a string of "best moves" even relative amateurs are stronger than their counterparts from years past.

Although the players of the past would face an uphill battle against the current top players they'd by no means be entirely out of their depth. There is also the question of fashion as many types of playing go in and out of style as time goes on. Some openings become more popular as a current top player plays it or fall aside the same way.

The answers given already have been quite excellent from a qualitative perspective and provide some great insight.

From a more quantitative perspective, I've found the work done by statistician/computer scientist/chess GM Ken Regan quite fascinating. He uses computer evaluations and statistical methods to calculate a "intrinsic performance rating" based on the strength of moves played.

It's hard to argue with the numbers with regards to the increasing strength of players over time. For example:

https://i.imgur.com/kvKdwLx.png

For reference, almost all of the top level events today have an IRR above 2700. Most of the world championship matches are pushing 2800.

His other papers on chess (found here) are also quite interesting and I highly recommend them.