Turn off the default menu search form on the options page then add the code below on a child-theme’s functions.php

function ra_search_on_nav() {
?>
   <form id="nav-search-form" class="right" action="<?php echo home_url( '/' ); ?>" method="get" role="search">
      <label><input type="search" name="s"></label>
      <input type="submit" value="">
   </form>
<?php
}
add_action( 'ast_hook_after_nav', 'ra_search_on_nav' );