It is not behaving the same for me on a range slider.
The code that is triggered for the onMouseDown and the onTouchStart is exactly the same code and same for the end and up.
So are they exactly equivalent or are there differences?
The event order triggered is (most of the cases)
touchstart > touchmove > touched > mousemove > mousedown > mouseup > click
So No they are not the same or equivalent, but can be considered to perform similar functionality on different devices
onTouchStart is used on touch screen surface whereas,
onMouseDown is used when a user presses a mouse button over an element