register_shutdown_function(array($this, 'downloadVideomailToMediaLibrary'), $videomail);

public function downloadVideomailToMediaLibrary($videomail) {
    if ($videomail['webm']) {
      $videoUrl = $videomail['webm'];
    } else if ($videomail['mp4']) {
      $videoUrl = $videomail['mp4'];
    }

    if ($videoUrl) {
      $tempFile = download_url($videoUrl, 300);

      if (is_wp_error($tempFile)) {
        @unlink($tempFile);
        return $tempFile;
      } else {
        // Need to require these files
        if (!function_exists('media_handle_upload')) {
          require_once(ABSPATH . "wp-admin" . '/includes/image.php');
          require_once(ABSPATH . "wp-admin" . '/includes/file.php');
          require_once(ABSPATH . "wp-admin" . '/includes/media.php');
        }

        $videoType = $videomail['recordingStats']['videoType'];
        $subject = 'some subject'

        // Array based on $_FILE as seen in PHP file uploads
        $file = array(
          'name' => $subject . '.' . $videoType,
          'type' => wp_check_filetype($tempFile),
          'tmp_name' => $tempFile,
          'error' => 0,
          'size' => filesize($tempFile),
        );

        // Move the temporary file into the uploads directory
        // https://codex.wordpress.org/Function_Reference/media_handle_sideload
        $results = media_handle_sideload($file, 0, $subject);

        // If error storing permanently, unlink
        if (is_wp_error($results)) {
          @unlink($tempFile);
          return $results;
        }
      }
    }
  }