Yes... and no.
What you're really doing by moving the thermostat is ensuring that the AC (or heater) stays on until the temp at the thermostat reaches the set temp. If the Thermostat is upstairs, the AC will remain on until the upstairs is 74 (if that's your set temp), regardless of what temp the downstairs is. The downstairs temp may well be 70, or 65, or anything else, because as you noted heat rises so the hottest part of the house will be upstairs. By the time the upstairs gets down to 74, the downstairs - which started at a lower temp - will be colder than 74. Worse, as the day goes on with a really hot day, the upstairs heats up more and faster than the downstairs, so the AC will keep kicking on to keep combatting that heat, which can have a cumulative effect on the downstairs. Let's say in the mid morning the AC runs a little and gets the upstairs to 74, but the downstairs is 70. Throughout the day, the upstairs will heat up more than the downstairs, so the AC will run enough to evacuate the heat upstairs. But there isn't the same amount of heat to evacuate downstairs. So by the end of the day, the upstairs may still be 74, but the downstairs might be 65.
The converse is true in winter. With the therm downstairs, it will heat both the bottom and top floors but the heat rises - by the time the downstairs reaches 74, the upstairs will be noticeably more warm, maybe 80 degrees.
Moving the thermostat isn't going to do anything for efficiency, nor will it even out the temps between your lower and upper floors. The only thing it would really accomplish is to allow you to set the thermostat to the minimum (in the winter) or maximum (in the summer) temp you want any floor in your house to be at. It would then get that floor to that temp, but the other floor will be lower or higher temp depending on time of year.
If you really want to solve the problem, what you would need is to consult with an hvac company about what products currently exist on the market that can be used to retrofit an existing heat pump system (I assume you have a heat pump if you have central heating and cooling) and make it a multizone system. This way, there would be a thermostat both upstairs and downstairs, and there are either motorized dampers or inflatable ballons that act as dampers in the ducts the partially or fully block airflow to individual ducts and registers based on the need for heating or cooling - if the upstairs is at the 74 set temp but the downstairs is not, the dampers will block off airflow to the upstairs ducts while leaving the system running until the downstairs is at the set temp as well, then shut off.