Some of the main problems with mobile synchronization of personal information are battery life and temporary loss of connection.
Thus, the usual way to do what you describe is to force the server to process most of the complex logic and multiple data sources to create a dataset that needs to be synchronized, and then have a proprietary protocol between the server and the client to mirror only that dataset .
In fact, the connection to the server will always be initiated by the client, regardless of how many people talk about push email. Your client application may have a user option so that the phone remains on the network, as far as network conditions permit. The server can respond to the connection by automatically sending the latest data necessary to synchronize with the client.
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