In my case, I have a scrollable body and a scrollable floating menu over it. Both have to be scrollable, but I had to prevent body scrolling when "floating menu" (position:fixed) received touch events and was scrolling and it reached top or bottom. By default browser then started to scroll the body.
I really liked jimmont's answer, but unfortunatelly it did not work well on all devices and browsers, especially with a fast and long swipe.
I ended up using MOMENTUM SCROLLING USING JQUERY (hnldesign.nl) on floating menu, which prevents default browser scrolling and then animates scrolling itself. I include that code here for completeness:
/**
 * jQuery inertial Scroller v1.5
 * (c)2013 hnldesign.nl
 * This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/.
 **/
/*jslint browser: true*/
/*global $, jQuery*/
/* SETTINGS */
var i_v = {
    i_touchlistener     : '.inertialScroll',         // element to monitor for touches, set to null to use document. Otherwise use quotes. Eg. '.myElement'. Note: if the finger leaves this listener while still touching, movement is stopped.
    i_scrollElement     : '.inertialScroll',         // element (class) to be scrolled on touch movement
    i_duration          : window.innerHeight * 1.5, // (ms) duration of the inertial scrolling simulation. Devices with larger screens take longer durations (phone vs tablet is around 500ms vs 1500ms). This is a fixed value and does not influence speed and amount of momentum.
    i_speedLimit        : 1.2,                      // set maximum speed. Higher values will allow faster scroll (which comes down to a bigger offset for the duration of the momentum scroll) note: touch motion determines actual speed, this is just a limit.
    i_handleY           : true,                     // should scroller handle vertical movement on element?
    i_handleX           : true,                     // should scroller handle horizontal movement on element?
    i_moveThreshold     : 100,                      // (ms) determines if a swipe occurred: time between last updated movement @ touchmove and time @ touchend, if smaller than this value, trigger inertial scrolling
    i_offsetThreshold   : 30,                       // (px) determines, together with i_offsetThreshold if a swipe occurred: if calculated offset is above this threshold
    i_startThreshold    : 5,                        // (px) how many pixels finger needs to move before a direction (horizontal or vertical) is chosen. This will make the direction detection more accurate, but can introduce a delay when starting the swipe if set too high
    i_acceleration      : 0.5,                      // increase the multiplier by this value, each time the user swipes again when still scrolling. The multiplier is used to multiply the offset. Set to 0 to disable.
    i_accelerationT     : 250                       // (ms) time between successive swipes that determines if the multiplier is increased (if lower than this value)
};
/* stop editing here */
//set some required vars
i_v.i_time  = {};
i_v.i_elem  = null;
i_v.i_elemH = null;
i_v.i_elemW = null;
i_v.multiplier = 1;
// Define easing function. This is based on a quartic 'out' curve. You can generate your own at http://www.timotheegroleau.com/Flash/experiments/easing_function_generator.htm
if ($.easing.hnlinertial === undefined) {
    $.easing.hnlinertial = function (x, t, b, c, d) {
        "use strict";
        var ts = (t /= d) * t, tc = ts * t;
        return b + c * (-1 * ts * ts + 4 * tc + -6 * ts + 4 * t);
    };
}
$(i_v.i_touchlistener || document)
    .on('touchstart touchmove touchend', function (e) {
        "use strict";
        //prevent default scrolling
        e.preventDefault();
        //store timeStamp for this event
        i_v.i_time[e.type]  = e.timeStamp;
    })
    .on('touchstart', function (e) {
        "use strict";
        this.tarElem = $(e.target);
        this.elemNew = this.tarElem.closest(i_v.i_scrollElement).length > 0 ? this.tarElem.closest(i_v.i_scrollElement) : $(i_v.i_scrollElement).eq(0);
        //dupecheck, optimizes code a bit for when the element selected is still the same as last time
        this.sameElement = i_v.i_elem ? i_v.i_elem[0] == this.elemNew[0] : false;
        //no need to redo these if element is unchanged
        if (!this.sameElement) {
            //set the element to scroll
            i_v.i_elem = this.elemNew;
            //get dimensions
            i_v.i_elemH = i_v.i_elem.innerHeight();
            i_v.i_elemW = i_v.i_elem.innerWidth();
            //check element for applicable overflows and reevaluate settings
            this.i_scrollableY      = !!((i_v.i_elemH < i_v.i_elem.prop('scrollHeight') && i_v.i_handleY));
            this.i_scrollableX    = !!((i_v.i_elemW < i_v.i_elem.prop('scrollWidth') && i_v.i_handleX));
        }
        //get coordinates of touch event
        this.pageY      = e.originalEvent.touches[0].pageY;
        this.pageX      = e.originalEvent.touches[0].pageX;
        if (i_v.i_elem.is(':animated') && (i_v.i_time.touchstart - i_v.i_time.touchend) < i_v.i_accelerationT) {
            //user swiped while still animating, increase the multiplier for the offset
            i_v.multiplier += i_v.i_acceleration;
        } else {
            //else reset multiplier
            i_v.multiplier = 1;
        }
        i_v.i_elem
            //stop any animations still running on element (this enables 'tap to stop')
            .stop(true, false)
            //store current scroll positions of element
            .data('scrollTop', i_v.i_elem.scrollTop())
            .data('scrollLeft', i_v.i_elem.scrollLeft());
    })
    .on('touchmove', function (e) {
        "use strict";
        //check if startThreshold is met
        this.go = (Math.abs(this.pageX - e.originalEvent.touches[0].pageX) > i_v.i_startThreshold || Math.abs(this.pageY - e.originalEvent.touches[0].pageY) > i_v.i_startThreshold);
    })
    .on('touchmove touchend', function (e) {
        "use strict";
        //check if startThreshold is met
        if (this.go) {
            //set animpar1 to be array
            this.animPar1 = {};
            //handle events
            switch (e.type) {
            case 'touchmove':
                this.vertical       = Math.abs(this.pageX - e.originalEvent.touches[0].pageX) < Math.abs(this.pageY - e.originalEvent.touches[0].pageY); //find out in which direction we are scrolling
                this.distance       = this.vertical ? this.pageY - e.originalEvent.touches[0].pageY : this.pageX - e.originalEvent.touches[0].pageX; //determine distance between touches
                this.acc            = Math.abs(this.distance / (i_v.i_time.touchmove - i_v.i_time.touchstart)); //calculate acceleration during movement (crucial)
                //determine which property to animate, reset animProp first for when no criteria is matched
                this.animProp       = null;
                if (this.vertical && this.i_scrollableY) { this.animProp = 'scrollTop'; } else if (!this.vertical && this.i_scrollableX) { this.animProp = 'scrollLeft'; }
                //set animation parameters
                if (this.animProp) { this.animPar1[this.animProp] = i_v.i_elem.data(this.animProp) + this.distance; }
                this.animPar2       = { duration: 0 };
                break;
            case 'touchend':
                this.touchTime      = i_v.i_time.touchend - i_v.i_time.touchmove; //calculate touchtime: the time between release and last movement
                this.i_maxOffset    = (this.vertical ? i_v.i_elemH : i_v.i_elemW) * i_v.i_speedLimit; //(re)calculate max offset
                //calculate the offset (the extra pixels for the momentum effect
                this.offset         = Math.pow(this.acc, 2) * (this.vertical ? i_v.i_elemH : i_v.i_elemW);
                this.offset         = (this.offset > this.i_maxOffset) ? this.i_maxOffset : this.offset;
                this.offset         = (this.distance < 0) ? -i_v.multiplier * this.offset : i_v.multiplier * this.offset;
                //if the touchtime is low enough, the offset is not null and the offset is above the offsetThreshold, (re)set the animation parameters to include momentum
                if ((this.touchTime < i_v.i_moveThreshold) && this.offset !== 0 && Math.abs(this.offset) > (i_v.i_offsetThreshold)) {
                    if (this.animProp) { this.animPar1[this.animProp] = i_v.i_elem.data(this.animProp) + this.distance + this.offset; }
                    this.animPar2   = { duration: i_v.i_duration, easing : 'hnlinertial', complete: function () {
                        //reset multiplier
                        i_v.multiplier = 1;
                    }};
                }
                break;
            }
            // run the animation on the element
            if ((this.i_scrollableY || this.i_scrollableX) && this.animProp) {
                i_v.i_elem.stop(true, false).animate(this.animPar1, this.animPar2);
            }
        }
    });
Another observation: I also tried various combinations of e.stopPropagation() on menu div and e.preventDefault() on window/body at touchmove event, but without success, I only managed to prevent scrolling I wanted and not scrolling I did not want. I also tried to have a div over whole document, with z-index between document and menu, visible only between touchstart and touchend, but it did not receive touchmove event (because it was under menu div).