Jonathan, if you have a log(ulog) that exhibits the pause would be great, as it'll show which module is causing the issue. Since you mention local_position, there could be multiple possibilities, especially if you're fusioning GPS and vio data. Based on one's setup, the local position NED calculation may use other sensors and not just vio.
To briefly check if vio is the source of the problem, you can look at Odometry topic (directly coupled to voxl vio). If vio has an issue, you'll see the pause, it goto zero or freeze. If not, it's likely a PX4 tuning issue. Nonetheless the ulog will explain what is happening.