The issue I was having was with some conflicting code I had in my functions file (see below). I usually use this code to remove width/height attributes from being hard-coded into the element in the content area. Removing these somehow appears to mess up the new responsive image implementation in WordPress though. As soon as I removed this code, the responsive images appear to be working as expected.
Conflicting Code
// Remove Width & Height Attributes from Inserted Images
add_filter( 'post_thumbnail_html', 'remove_width_attribute', 10 );
add_filter( 'image_send_to_editor', 'remove_width_attribute', 10 );
function remove_width_attribute( $html ) {
$html = preg_replace( '/(width|height)="\d*"\s/', "", $html );
return $html;
}