eSeGeCe has a set of components that implement the WebSocket protocol in Delphi XE5 Firemonkey on IOS and Android. WebSockets are what is implemented in web browsers with HTML5. There are both client and server components included in the suite and it has four built in sub protocols which are JSON-RPC 2.0, Datasets, WebRTC and WAMP. It is a commercial component suite but the components can be used for free even for commercial use. If you want source code you can purchase that. SSL and TLS are also supported through the OpenSSL libraries. There is a server and a client demo for Firemonkey and a few other demos also included and it also looks like Windows and OSX are supported platforms.
FMXExpress.com has over 700 tips and tricks for building cross platform apps on Android, IOS, OSX, Windows, and Linux with a single codebase and single UI using the FireMonkey framework found in Delphi and C++Builder.
FMXExpress covers top cross platform secrets for FireMonkey in C++Builder and Delphi XE5, XE6, XE7, XE8, Seattle, Berlin, Tokyo, Rio, and Sydney.