Just like applications, games will be migrated to the browser as well. In order to understand the forces that will make this transition happen, we need to understand what makes the Internet, the browsers and websites so successful in the first place, and why is that relevant in the context of games.
Bit of History
There is always a reason why a trend develops and decays over time. It's more than just excitement then later on boredom. In the beginning, all the application were developed for the mainframes first, because at the time no single person could afford a computer. Then the PC came along and everything got rewritten for that platform. Then just recently, internet speed, hardware and browsers become fast enough to run the apps we use in the office to trigger another wave of application migration, back to the cloud. Think about Microsoft Office Online. Improvements on the hardware level will disrupt trends build on top of it.
The Bias
When a new technology comes along, people tend to focus on the downsides only and argue that the new tech never going to be as good as the old one. The reasonable question would be, what are the downsides and what are the upsides and what the new technology allows me to do and can I live with the downsides. Change is difficult, therefore people tend to stick to the old tech. New customers who are exposed to competing products the first time, evaluate the old and the new tech objectively and then make a decision.
People tend to confuse the principles that brought a trend to live with the current state of those trends. They say: yes, but at the moment X can’t do what Y can. A set of features was used to create a popular app, service or game, a different set of features will produce a different product. The new product doesn’t have to be superior in every way in order to succeed.
Not as Black and White as we’d like it to be
When PC was introduced it wasn’t as fast as consoles at the time. People argued that it's not good for gaming. Today it's the other way around. Nintendo Switch was released with 1 teraflop performance on a market where the competition was PS 4 Pro - 4.2 teraflop - and XBox One X - 6 teraflop and become one the fastest selling consoles of all times. Clearly, there was more to the story than just performance. After all, why do even consoles exist when they are inferior to the PC.
So what are the upsides of gaming in the browser and what do we have to put up with?
Upsides:
- Instant Gaming - fast as loading up a website
- Cross play - No need to convince your friends to buy the same console to play the same games together.
- Seamless Updates - No delayed gaming sessions because a new 30GB update just got released.
- Low Resistance - When something is only one click away, it's more likely to give it a try.
- One Standard - Porting Games is unnecessary. If you have a browser, you have access to the game.
Downsides:
- Limited Graphics - Web standards are behind the cutting edge APIs available on other platforms.
- Fast Internet Required - You need a reasonably good internet speed. E.g.: 4G on your phone.
- Deviation from Standards - Some browser vendors make it difficult for devs to use the same codebase.
Conclusion
We probably going to see a form of merger in the long run. Apps will merge with websites and the same goes for games, and they partly already did on mobile devices. It's not the question of better or worse, or whether it will provide a better experience for everybody or not. It's more about what will become mainstream? When most people sit down to play a game, what device they will reach out to and what platform will deliver most games?
Feel free to disagree and leave a comment below if you have something to say 
I believe that games will not become browser based, inevitably. Here's why:
LIBRARIES!!!
With the push for higher end graphics, and the want of the consumer to have these higher quality graphics, I think that unless browsers are able to support higher end graphics libraries (like we are seeing with WebGL, but now people are making the move to Vulkan. Will we have Web-kan?) that we won't see a switch to browsers anytime soon, at least for AAA games. Not to mention that of the tools available, there is a limited subset that is available in browsers today. From what I know, there isn't a 3D audio library for browsers, nor is there any ability to add scripting support (I suppose you could add it in the back-end, but that would be a little bit too much for the browser to handle with those get and post requests.)
PERFORMANCE!!!
Another thing to mention is the ability for a native program to access the memory specified by the OS, rather than using browser memory. This allows programmers to have access to a larger array of memory, and speeds up memory access, than just requesting the browser to request memory from the OS. Also, with a program like a game, it can take advantage of the OS scheduler and be put into the correct priority that the game needs to, rather than being stuck in a browser and doing goodness-knows-what to the scheduler.
I am biased in this regard though: I hate browsers using up almost all of my computers processing time and memory.