WordPress Multisite: Password Reset on a Subsite.

functions.php

/**
 * Password reset on sub site (1 of 4)
 * Replace login page "Lost Password?" urls.
 *
 * @param string $lostpassword_url The URL for retrieving a lost password.
 * @param string $redirect The path to redirect to.
 *
 * @return string
 *
 * @since 1.0.0
 */
function custom_lostpassword_url( $lostpassword_url, $redirect ) {
	$use_url = false;
	$args    = array( 'action' => 'lostpassword' );

	if ( ! empty( $redirect ) ) {
		$args['redirect_to'] = $redirect;
	}

	if ( $use_url ) {
		return esc_url( add_query_arg( $args, $lostpassword_url ) );
	}

	return esc_url( add_query_arg( $args, site_url( 'wp-login.php' ) ) );
}
add_filter( 'lostpassword_url', 'custom_lostpassword_url', 10, 2 );

/**
 * Password reset on sub site (2 of 4)
 * Replace other "unknown" password reset urls.
 *
 * @param string $url The complete network site URL including scheme and path.
 *
 * @return string
 *
 * @since 1.0.0
 */
function replace_lostpassword_urls( $url ) {
	if ( stripos( $url, 'action=lostpassword' ) !== false ) {
		return site_url( 'wp-login.php?action=lostpassword' );
	}

	if ( stripos( $url, 'action=resetpass' ) !== false ) {
		return site_url( 'wp-login.php?action=resetpass' );
	}

	return $url;
}
add_filter( 'network_site_url', 'replace_lostpassword_urls', 10, 3 );

/**
 * Password reset on sub site (3 of 4)
 * Fixes the URLs in emails that are sent.
 *
 * @param string $message Default mail message.
 *
 * @return string
 *
 * @since 1.0.0
 */
function retrieve_password_message_urls( $message ) {
	return str_replace( get_site_url( 1 ), get_site_url(), $message );
}
add_filter( 'retrieve_password_message', 'retrieve_password_message_urls' );

/**
 * Password reset on sub site (4 of 4)
 * Fixes the title in emails that are sent.
 *
 * @return string
 *
 * @since 1.0.0
 */
function custom_retrieve_password_title() {
	return sprintf( __( '[%s] Password Reset' ), wp_specialchars_decode( get_option( 'blogname' ), ENT_QUOTES ) );
}
add_filter( 'retrieve_password_title', 'custom_retrieve_password_title' );