ajax.js

;( function( $, obj ) {
  var uno = function( e ) {
		e.preventDefault();

		var $this = $( this );

		$.ajax( {
			type:  "POST",
			url:   obj.ajaxurl
				       + "?action=" + obj.action
				       + "&" + obj.nonce_query_arg + "=" + obj.nonce,
			nonce: obj.nonce,
			data:  {
				obj: obj
			},

			beforeSend: function( jqXHR, settings ) {
				console.log( 'beforeSend' );
			}
		} ).done( function( data, textStatus, jqXHR ) {
				console.log( textStatus );
			} ).fail( function( jqXHR, textStatus, errorThrown ) {
				console.log( textStatus );
				console.log( errorThrown );
			} );
	};

	$.each( $( ".uno" ), function( index, el ) {
		$( el ).on( 'click', uno );
	} );

	$.each( $( ".due" ), function( index, el ) {
		$( el ).on( 'click', uno );
	} );

} )( jQuery, user_actions_obj || {} );

bootstrap.php

<?php
/* Plugin Name: AJAX callback test */

# ======= ADD MARKUP TO LOOP

add_action( 'plugins_loaded', 'setup_test_plugin' );
function setup_test_plugin()
{
  require_once 'classes.php';

	foreach ( array( 'testUNO', 'testDUE' ) as $callback )
		add_filter( 'the_content', $callback );
}

function testUNO()
{
	print apply_filters(
		'user_action_markup_uno',
		get_the_ID()
	);
}
function testDUE()
{
	print apply_filters(
		'user_action_markup_due',
		get_the_ID()
	);
}

classes.php

<?php
# ======= PARENT CLASS

abstract class AbstractBaseClass
{
  public $file_name = 'ajax';
	public $handle    = 'user_actions';
	public $action    = '';

	public $nonce_val       = 'user_actions';
	public $nonce_query_arg = 'user_actions_nonce';

	public $markup_filter = '';

	public $classes = array(
		'add'      => 'theme-add-link',
		'fav'      => 'theme-fav-link',
		'unfav'    => 'theme-unfav-link',
		'download' => 'theme-download-link',

		'wait'     => 'theme-wait-state',
		'error'    => 'theme-error-state',
		'done'     => 'theme-done-state',
	);

	public function __construct()
	{
		$this->action = "{$this->handle}_action";

		add_action( 'wp_enqueue_scripts', array( $this, 'scripts' ) );

		$this->markup_filter = $this->set_markup_filter_name();
		add_filter( $this->markup_filter, array( $this, 'get_markup' ) );
	}

	public function scripts()
	{
		wp_enqueue_script(
			$this->handle,
			plugin_dir_url( __FILE__ )."/{$this->file_name}.js",
			array( 'jquery' ),
			null,
			true
		);
		wp_localize_script(
			$this->handle,
			"{$this->handle}_obj",
			array(
				'ajaxurl'         => admin_url( 'admin-ajax.php' ),
				'nonce_query_arg' => $this->nonce_query_arg,
				'nonce'           => wp_create_nonce( $this->nonce_val ),
				'action'          => $this->action,

				'classes'         => $this->classes,
			)
		);
	}

	abstract function set_markup_filter_name();

	abstract function ajax_cb( $data );

        public function ajax_exit_cb( $data ){
             exit();
        }

	public function check_ajax()
	{
		check_ajax_referer( $this->nonce_val, $this->nonce_query_arg );
		if (
			! isset( $_POST['obj']['action'] )
			OR $this->action !== $_POST['obj']['action']
		)
			exit( -1 );
	}

	abstract function get_markup( $post_id );
}

# ======== EXTENDING CLASSES


add_action( 'plugins_loaded', array( 'ChildClassUNO', 'init' ), 30 );
class ChildClassUNO extends AbstractBaseClass
{
	protected static $instance = null;

	public static function init()
	{
		null === self::$instance AND self::$instance = new self;
		return self::$instance;
	}

	public function __construct()
	{
		parent::__construct();

		add_action( "wp_ajax_{$this->action}", array( $this, 'ajax_cb' ) );
		add_action( "wp_ajax_nopriv_{$this->action}", array( $this, 'ajax_cb' ) );
		add_action( "wp_ajax_{$this->action}", array( $this, 'ajax_exit_cb' ), 11 );
		add_action( "wp_ajax_nopriv_{$this->action}", array( $this, 'ajax_exit_cb' ), 11 );
	}

	public function set_markup_filter_name()
	{
		return 'user_action_markup_uno';
	}

	public function get_markup( $post_id )
	{
		return sprintf(
			'<a href="#" class="uno" id="uno-%s"> UNO </a>',
			$post_id
		);
	}

	public function ajax_cb( $data )
	{
		print __CLASS__;
		$this->check_ajax();
	}
}

add_action( 'plugins_loaded', array( 'ChildClassDUE', 'init' ), 30 );
class ChildClassDUE extends AbstractBaseClass
{
	protected static $instance = null;

	public static function init()
	{
		null === self::$instance AND self::$instance = new self;
		return self::$instance;
	}

	public function __construct()
	{
		parent::__construct();

		add_action( "wp_ajax_{$this->action}", array( $this, 'ajax_cb' ) );
		add_action( "wp_ajax_nopriv_{$this->action}", array( $this, 'ajax_cb' ) );
	}

	public function set_markup_filter_name()
	{
		return 'user_action_markup_due';
	}

	public function get_markup( $post_id )
	{
		return sprintf(
			'<a href="#" class="due" id="due-%s"> DUE </a>',
			$post_id
		);
	}

	public function ajax_cb( $data )
	{
		print __CLASS__;
		$this->check_ajax();
	}
}

Removed exit() calls from child classes. Moved it into base class which calls it after other (child class) callbacks have been called.

Given the ajax_exit_cb() method a lower priority (higher number) just for good measure. Must run after all child class callbacks on the same hook.