Erin Stellato shows us that Extended Events and Profiler both use local temp storage:
Depending on what events you have configured for Profiler, your filter(s), the workload, and how long you run Profiler, you could generate more events than the UI can handle. Therefore, they’ll start buffering to the User TMP location. If you’re not paying attention, you can fill up the C: drive. This can cause applications (including SQL Server) to generate errors or stop working entirely. Not good.
Reference: https://msdn.microsoft.com/en-us/library/ms174203.aspx
Now, back to the original question. Does the same problem exist for Extended Events? Only if you’re using the Live Data Viewer. After you have an event session created (you can just use system_health for this example), within Management Studio, go to Management | Extended Events | Sessions, select the session and right-click and select Watch Live Data
This is one of those things you hardly think about, but it makes sense: that data’s got to be stored somewhere if things are moving too fast for the app.