How does the stomach handle continuous eating?
First of all, the stomach releases the digested food continuously. Simply put, during digestion the pylorus regularly opens a bit so that small food particles (&lt; 1-2mm) are able to leave the stomach – it's not a batch process (see also this question).

However, because humans tend to ingest a lot of indigestible materials (e.g. bones), there is a need to pass the larger indigestible particles – and this process only happens after all digestible food has left the stomach and thus is "reset" every time you ingest new digestible food.

So if you would continuously ingest food that contains indigestible parts (e.g. mice or other small mammals), the larger bones of those mice would accumulate until you take a longer break and the stomach could complete it's digestion – but for normal food, this does not matter since it will be broken down enough to leave your stomach continuously.



Source: According to Estimation of gastric residence time of the Heidelberg capsule in humans: Effect of varying food composition, 1985 (PDF), the  interdigestive migrating myoelectric complex (IMMC) is responsible for the emptying of of large undigested particles:


  Recently, Hinder and Kelly (1), using radiolabeled techniques, demonstrated differences in the gastric emptying patterns of a liquid, a digestible solid, and an indigestible solid in dogs. Based on such experiments, it has been suggested that the emptying of large (>1 mm) indigestible objects from the stomach
  is dependent on the interdigestive migrating myoelectric complex (IMMC)


Furthermore the study shows that the IMMC gets triggered only after all digestible matter has leaved the stomach:


  The longer GRT of the Heidelberg capsule compared with the t1/2 of the 99mTC-DTPA is consistent with the finding that large nondigestible solids are emptied by the IMMC once all of the digestible materials have passed through the pylorus into the duodenum


and


  the interdigestive migrating myoelectric complex can be markedly delayed by frequent feedings with solids, and the interdigestive migrating myoelectric complex is delayed by both liquid and solid meals


and


  Feeding has been shown to interrupt the IMMC; resumption of myoelectric activity is necessary for the passage of large nondigestible particles such as the Heidelberg capsule