Dragable infinite loop carousel, performance issues on dragging large number (25+) of slides
From a once over, some performance tips:


At least on some versions of Chrome, parseInt(string,10); is slower than a number of alternatives, my preference is +string
It is faster to call Math.abs(string) than Math.abs(parseInt(string,10))
Each object property lookup has a cost, so consider var base = this.protected; since all your object accesses are on `base.protected'
Be wary of ternary operators, a regular if tends to be faster
Your code has actually a number of little inefficiencies

You will capture var MousePosition = GetEventCoordinates(ev); but then later you still call locals.NewSliderLeft = GetEventCoordinates(ev).x - locals.DragStartTouchOffsetX-locals.PopAdjustment;
You store locals.NewSliderLeft but you use it only once in the assignment here: base.protected.$Slider.css('left',locals.NewSliderLeft);, I would skip storing this value in locals altogether.

From a style perspective, I would suggest that you use more spaces and fewer parentheses so var MouseMoveDistance = (MousePosition.x-locals.StartingMousePosition.x); becomes 
var MouseMoveDistance = MousePosition.x - locals.StartingMousePosition.x;
Also prefixing in general is discouraged (oEdgeCounts)
From a DRY perspective, the block of if(oEdgeCounts.left&lt;1 &amp;&amp; MoveDirection==='&gt;'){ and if(oEdgeCounts.right&lt;1 &amp;&amp; MoveDirection==='&lt;'){ is pretty much copy pasted, you should refactor that


All in all, your code is quite readable and maintainable and I dont see immediately where the culprit is. I fully endorse the opion of Doug Gale to use the Chrome developer tools to find the culprit.