You cannot to an “IN” query on a relationship field.
see section 3 of this doc https://www.advancedcustomfields.com/resources/query-posts-custom-fields/
You need to do
$meta_query = array(
'relation' => 'OR',
array(
'key' => 'field_name',
'value' => '"'.$first_id.'"',
'compare' => 'LIKE'
),
array(
'key' => 'field_name',
'value' => '"'.$second_id.'"',
'compare' => 'LIKE'
),
// etc..
);