Global Site Search

Yes – indeed it can be used to search network-wide. Upon the initial index step you must use the –network-wide WP CLI command.

So the command would go:
wp elasticpress index --setup --network-wide
https://github.com/10up/ElasticPress#multisite-cross-site-search

This will ensure that all your subsites are correctly indexed.

You must also define the sites parameter to all in your WP_Query.

new WP_Query( array(
    's'     => 'search phrase',
    'sites' => 'all',
) );

https://github.com/10up/ElasticPress#supported-wp_query-parameters