Every year we experience spring, summer, autumn and winter, caused by the position of the Sun. On a globe model the sun appears to be travelling from 23 degree latitude north to 23 degree latitude south. But how would this work on a flat earth model?
How can the speed vary?
On a flat plane the earth is the centre of the universe with the Sun revolving at a constant 24 hours per day. In this case the easiest way is to consider that the north-pole is actually a pole and at its highest point a beam is holding the sun in position. As the pole rotates in exactly 24 hours then the beam must be able to lower the sun as the seasons change. As the constant is 24 hours, then the distance increases with the lowering of the Sun resulting in a greater speed. So at 23 degrees north the Sun actually would go the slowest and at 23 degrees south the fastest.
How fast does the Sun go?
To calculate how fast the sun would go, we need to determine the radius of the Sun's path at 23 degree north, south and at the equator. This I will do based on the know radius of the globe aka 6371 km. As the earth is pushed into a flat earth then the distance from the north-pole to the equator would be 6371*2*pi/360*90 = 10007 km resulting in a circumference of 10007*2*pi = 62831 km giving a speed at the equator of the sun of 62831/24 = 2617 km/hour. At 23 degree north this means a radius of 7450 km, a circumference of 46810 km and a speed of 1950 km/hour. At 23 degree south this means a radius of 12565 km, a circumference of 78948 km and a speed of 3290 km/hour. So if the earth is a flat plane than the Sun must have different speeds!
PS: the following impression isn't how the world looks like, it is a simple way to show how the speed could change with a change of position based on a continuous 24 hour time-frame.