Views Slideshow Cycle doesn't re-initialize properly after an AJAX content update
I would expect the behaviors mechanism to somehow kill the old scripts
  and settings and replace them with new ones upon processing an AJAX
  response but it doesn't seem to happen and I can't find any
  documentation on this particular aspect.


The D7 Managing Javascript docs explains this well:


  Drupal.behaviors will often be called multiple times on a page. For
  example, every time a form performs some Ajax operation, all Drupal
  behaviors will be executed again after page load, in order to attach
  any relevant JavaScript to the newly loaded elements. This can have
  the undesired affect of applying JavaScript to elements each time
  Drupal behaviors are executed, resulting in the same code being
  applied multiple times. To ensure that the JavaScript is applied only
  once, we can use the jQuery $.once() function. This function will
  ensure that the code inside the function is not executed if it has
  already been executed for the given element.


The D8 Javascript Overview docs tells you how to use $.once() in D8. 

If your code is not recognizing the ajaxed in divs, then you need to attach the behaviors to the ajax wrapper, with drupal.attachBehaviors($('.ajax-wrapper'));

Ex:

$.ajax({
    url: '/some/url',
    type: 'GET',
    success: function (data) {
        var result = $(data).find('.some-wrapper').html();
        $('.ajax-wrapper').html(result);
        Drupal.attachBehaviors($('.ajax-wrapper'));

    }
});