List of Events with Multiple Dates: Only NEXT Date

I think that you have poor data design. The solution, in my opinion, is to alter how your data is saved. If you saved all of your dates under the same key name you could run a relatively simple query to get the next date.

$args = array(
'meta_query'    => array(
    'relation' => 'OR',
    array(
    'key'        => 'event_dates',
    'value'      => date("Y-m-d H:i:s",strtotime('today midnight')), // set
    'compare'    => '>',
    'type'       => 'DATE'
    ),
    ),
'posts_per_page'  => 30,
'post_type'     => 'performances'
);
$loop = new WP_Query( $args );
var_dump($loop->request);