Prices will stabilize across markets. If prices get comparatively cheaper on one market more purchases will occur leading to higher demands, more scarcity and more value. If prices were to get comparatively high more selling would occur on that market, more supply, less scarcity and less value.
The key point to take from this paragraph is that there is not nescesarily one person who is buying/selling on different markets to level the price. People who have money on the higher exchange will be more likely to sell, and people who already have money on the lower exchange will be more likely to buy.
You could try to do the arbitrage yourself, but between the fees and the time to move capital between exchanges, you will never make enough money for the effort to be worth it.