Test widget plugin for WordPress issue #27491: Widget Customizer: Dynamically-created inputs cause from to replace itself without event to trigger re-initialization https://core.trac.wordpress.org/ticket/27491

.gitmodules

[submodule "chosen"]
	path = chosen
	url = git@github.com:harvesthq/chosen.git

customize-widgets-dynamic-fields-test.js

/*global jQuery */

jQuery( function ( $ ) {

	function init( widget_el, is_cloned ) {

		// Clean up from the clone which lost all events and data
		if ( is_cloned ) {
			widget_el.find( '.chosen-container' ).remove();
		}

		widget_el.find( 'select.dynamic-field-chosen-select' ).chosen( { width: '100%' } );
	}

	/**
	 * @param {jQuery.Event} e
	 * @param {jQuery} widget_el
	 */
	function on_form_update( e, widget_el ) {
		if ( 'dynamic_fields_test' === widget_el.find( 'input[name="id_base"]' ).val() ) {
			init( widget_el, 'widget-added' === e.type );
		}
	}

	$( document ).on( 'widget-updated', on_form_update );
	$( document ).on( 'widget-added', on_form_update );

	$( '.widget:has(.dynamic-field-chosen-select)' ).each( function () {
		init( $( this ) );
	} );

} );

customize-widgets-dynamic-fields-test.php

<?php
/**
 * Plugin Name: Customize Widget Dynamic Fields Test
 * Description: For testing purposes.
 * Author: Weston Ruter, X-Team
 * Author URI: http://x-team.com/profile/weston-ruter/
 */

class Dynamic_Fields_Test_Widget extends WP_Widget {
	public $chosen_options;

	function __construct() {
		parent::__construct(
			'dynamic_fields_test',
			'Dynamic Fields test',
			array(
				'description' => 'Test customizing widgets with dynamic fields.',
			)
		);

		$this->chosen_options = array(
			'' => __( '(None)' ),
			'red' => __( 'Red' ),
			'green' => __( 'Green' ),
			'blue' => __( 'Blue' ),
		);
	}

	/**
	 * Front-end display of widget.
	 *
	 * @param array $args     Widget arguments.
	 * @param array $instance Saved values from database.
	 */
	public function widget( $args, $instance ) {
		$instance = wp_parse_args( $instance, $this->default_instance() );
		$title = apply_filters( 'widget_title', $instance['title'] );

		echo $args['before_widget']; // xss ok
		if ( ! empty( $title ) ) {
			echo $args['before_title'] . $title . $args['after_title']; // xss ok
		}
		echo '<pre>';
		echo esc_html( print_r( $instance, true ) );
		echo '</pre>';

		echo $args['after_widget']; // xss ok
	}

	/**
	 * Back-end widget form.
	 *
	 * @param array $instance Previously saved values from database.
	 * @return void
	 */
	public function form( $instance ) {

		wp_enqueue_style(
			'chosen',
			plugin_dir_url( __FILE__ ) . '/chosen/chosen.css'
		);
		wp_enqueue_script(
			'chosen',
			plugin_dir_url( __FILE__ ) . '/chosen/chosen.jquery.js',
			array( 'jquery' )
		);
		wp_enqueue_script(
			'dynamic-fields-test-widget',
			plugin_dir_url( __FILE__ ) . '/' . basename( __FILE__, '.php' ) . '.js',
			array( 'jquery', 'chosen' )
		);

		$instance = wp_parse_args( $instance, $this->default_instance() );
		?>
		<p>
			<label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:' ); ?></label>
			<input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo esc_attr( $instance['title'] ); ?>">
		</p>
		<p>
			<label for="<?php echo $this->get_field_id( 'chosen' ); ?>"><?php _e( 'Chosen:' ); ?></label>
			<select id="<?php echo $this->get_field_id( 'chosen' ); ?>" class="dynamic-field-chosen-select" name="<?php echo $this->get_field_name( 'chosen' ); ?>">
				<?php foreach ( $this->chosen_options as $value => $text ): ?>
					<option value="<?php echo esc_attr( $value ) ?>" <?php selected( $value, $instance['chosen'] ) ?>><?php echo esc_html( $text ) ?></option>
				<?php endforeach; ?>
			</select>
		</p>
		<p>
			<?php _e( 'Form loaded at: ' ) ?><?php echo date( 'H:m:s' ) ?>
		</p>
		<?php
	}

	/**
	 * @return array
	 */
	public function default_instance() {
		return array(
			'title' => '',
			'chosen' => '',
		);
	}

	/**
	 * Sanitize widget form values as they are saved.
	 *
	 * @see WP_Widget::update()
	 *
	 * @param array $new_instance Values just sent to be saved.
	 * @param array $old_instance Previously saved values from database.
	 *
	 * @return array Updated safe values to be saved.
	 */
	public function update( $new_instance = array(), $old_instance = array() ) {
		$instance = wp_parse_args( $new_instance, $this->default_instance() );
		$instance['title'] = strip_tags( $new_instance['title'] );
		if ( ! isset( $this->chosen_options[ $instance['chosen'] ] ) ) {
			$instance['chosen'] = '';
		}
		return $instance;
	}

}

add_action( 'widgets_init', function () {
	register_widget( 'Dynamic_Fields_Test_Widget' );
} );

I have the same problem but found this https://core.trac.wordpress.org/ticket/19675 yet still can not load the script then to drag the widget

add a new field in all widget but in the event you add it to a sidebar do not load me the script, specifically a "WpColorPicker' but I do not load, I only load when save but when dragging no longer works for me.

http://i.imgur.com/fv3xTcD.png