Using the new drawer menu is as simple as changing what was previously data-toggle="collapse" to data-toggle="drawer" and adding the .drawer class to the .navbar-header.
<div class="drawer navbar-header" data-toggle="drawer">
<button type="button" class="navbar-toggle" data-toggle="drawer" data-target=".navbar-responsive-collapse">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="#">Brand</a>
</div>