How do I interact with $title, $breadcrumbs, and $action_links within a process or preprocess function?
This is how I manipulate breadcrumbs in page preprocess function. You can actually do this somewhere else, where you think it fits according to your needs.

/**
 * Implements hook_preprocess_page().
 */
function extremes_preprocess_page(&amp;$variables) {
  // You can use preprocess hooks to modify the variables before they are passed
  // to the theme function or template file.

  //define breadcrumbs array
  $breadcrumb = array();
  $breadcrumb[] = l('SS','&lt;front&gt;');//add homepage first
  //if we are on one of my dynamic pages, we do this
  if (arg(0) == 'node' &amp;&amp; is_numeric(arg(1)) &amp;&amp; !empty($variables['node']) &amp;&amp; _is_dynamic_type($variables['node']-&gt;type)) { 
    if (!empty($variables['node']-&gt;field_term_reference)) {//we are gonna depend on a term ref field
      //get all parent terms
      $parents = array_reverse(taxonomy_get_parents_all($variables['node']-&gt;field_term_reference[LANGUAGE_NONE][0]['target_id']));
      //and add them to breadcrumb array
      foreach ($parents as $parent) {
        $breadcrumb[] = l($parent-&gt;name, 'taxonomy/term/' . $parent-&gt;tid);
      }
    }
    //set the breadcrumb
    drupal_set_breadcrumb($breadcrumb);
  }
  //if we are on a term listing page
  else if (arg(0) == 'taxonomy' &amp;&amp; is_numeric(arg(2))) {//you can also check here, which vocabulary you want
    //easier this time
    $parents = array_reverse(taxonomy_get_parents_all(arg(2)));
    foreach ($parents as $parent) {
      $breadcrumb[] = l($parent-&gt;name, 'taxonomy/term/' . $parent-&gt;tid);
    }
    if (arg(3) &amp;&amp; $type = taxonomy_term_machine_name_load(arg(3), 'vocab_name')) {
      $breadcrumb[] = l($type-&gt;name, request_path());
    }
    drupal_set_breadcrumb($breadcrumb);
  }

}


This sample uses taxonomy_machine_name.module which uses the same system with pathauto and is very useful for loading taxonomy terms from url if you set up correctly.

For title, you can go with drupal_set_title() and for action links hook_menu_local_tasks_alter().