For a simple answer: no, most mobile robots do not use variable transmissions. I believe it is far easier to just design a motor and gearbox combination that works well in your narrow range of desired speeds, than deal the extra mechanical complexity and cost of a variable transmission.
The long answer is that it depends on what you consider a mobile ...
This project does exactly that on an RC car. The author is a top competitor in the DIYRobocars community; it's the blue car in this video. He uses tachometry from the brushless motor, an IMU and visual odometry for localization. I don't know the code well enough to point you to any specific file.
Yes, brushless and grearbox is a normal usecase. Due to the popular application of using brushless motors on drone propellers other applications are harder to find. The keywords "Brushless Planetary Gear Motor" gives me plenty good search results.