For Android the limitation on the push notification size is 4k. If your messages are smaller than that, you can use push notifications for sending all the content.
However, delivery of push notifications is not guaranteed, so it's a better practice to use if as a means for the server to notify the app that new data is available, especially when it's not running in the foreground. The app then calls the server to retrieve the data.
When the app is in the foreground, you can poll the server periodically for new data.