The Dow Jones is just an index, a secondary calculation out of various share prices; it has nothing to do with where shares are traded.
Shares are often traded at many exchanges; each exchange has its own rules on what they trade, and globally well known shares like Apple or Alphabet might be traded on many different exchanges all over the world, where smaller and lesser known shares are often only traded on their 'home' country's exchange.
There is no hard rule. Consider it like car trades - you can buy a Toyota in many places, most towns have dealerships that trade them; whereas Lamborghinis have a very limited number of dealerships, and you might have to go quite some distance to find one.
Each exchange has its fixed trading times, so depending on which exchange deals with certain shares, those shares can be traded at different times. Back to the car example: if the Toyota dealer in your town opens at 5 am, you can buy a Toyota at 5 am.