This snippet maps mouse events and touch events onto one single event. This makes it easier in the code since you have to listen to only one event regardles whether it's desktop or mobile.

touchmouse.js

(function() {
    
    /* == GLOBAL DECLERATIONS == */
    TouchMouseEvent = {
        DOWN: "touchmousedown",
        UP: "touchmouseup",
        MOVE: "touchmousemove"
    }
   
    /* == EVENT LISTENERS == */
    var onMouseEvent = function(event) {
        var type;
        
        switch (event.type) {
            case "mousedown": type = TouchMouseEvent.DOWN; break;
            case "mouseup":   type = TouchMouseEvent.UP;   break;
            case "mousemove": type = TouchMouseEvent.MOVE; break;
            default: 
                return;
        }
        
        var touchMouseEvent = normalizeEvent(type, event, event.pageX, event.pageY);      
        $(event.target).trigger(touchMouseEvent); 
    }
    
    var onTouchEvent = function(event) {
        var type;
        
        switch (event.type) {
            case "touchstart": type = TouchMouseEvent.DOWN; break;
            case "touchend":   type = TouchMouseEvent.UP;   break;
            case "touchmove":  type = TouchMouseEvent.MOVE; break;
            default: 
                return;
        }
        
        var touch = event.originalEvent.touches[0];
        var touchMouseEvent;
        
        if (type == TouchMouseEvent.UP) 
            touchMouseEvent = normalizeEvent(type, event, null, null);
        else 
            touchMouseEvent = normalizeEvent(type, event, touch.pageX, touch.pageY);
        
        $(event.target).trigger(touchMouseEvent); 
    }
    
    /* == NORMALIZE == */
    var normalizeEvent = function(type, original, x, y) {
        return $.Event(type, {
            pageX: x,
            pageY: y,
            originalEvent: original
        });
    }
    
    /* == LISTEN TO ORIGINAL EVENT == */
    var jQueryDocument = $(document);
   
    if ("ontouchstart" in window) {
        jQueryDocument.on("touchstart", onTouchEvent);
        jQueryDocument.on("touchmove", onTouchEvent);
        jQueryDocument.on("touchend", onTouchEvent); 
    } else {
        jQueryDocument.on("mousedown", onMouseEvent);
        jQueryDocument.on("mouseup", onMouseEvent);
        jQueryDocument.on("mousemove", onMouseEvent);
    }
    
})();

How would I deal with things like targetTouches or the equivalent of that for mouse events?

Adding this piece, in the "onMouseEvent" listener, will remove right click:

if (event.which === 3) { return; }

There is a problem with touch notebooks. You can't use the "Click"-Event of the mouse because its detected as Touch-Only Device ("ontouchstart" in window).

Any suggesting how to overcome this problem?