An external tool with embed option works properly on desktop browsers, with the content showing inside of the Moodle theme, and the URL on the page is still that of Moodle.
However, the same link behaves differently on mobile browsers. The link would launch as if it were redirected, with the launch URL of the LTI tool showing in the address bar instead of our Moodle's URL, and the content filling the entire window. This happened on Safari and Chrome on the iPad Air, and the "default web browser" and Chrome on Galaxy Tab 10 and Galaxy Nexus.
We tested the browsers using just a generic page with iframe and object, and they all work as wanted, with the iframe content staying inside the iframe
Has anyone experienced this? Is there anything in Moodle's External Tool LTI implementation that could have caused this?