1. A computer-implemented method for remote session browsing comprising: obtaining, by a network computing provider comprising one or more server computing devices, a request for a persistent browsing context from a first client-side browser application executing on a first client computing device; initializing a server-side browser application, separate from the first client-side browser application, on a server computing device of the network computing provider in response to the request for the persistent browsing context; obtaining content requested by the first client-side browser application, the content comprising a network resource referencing one or more embedded resources, wherein the content requested by the first client-side browser application is to be processed by a plurality of processing actions to the render the content, wherein the server-side browser application is configured to perform each of the plurality of processing actions to render the content, and wherein the first client-side browser application is also configured to perform each of the plurality of processing actions to render the content; generating, by the server-side browser application, a first processing result by performing at least a portion of the plurality of processing actions to render the content, wherein the first processing result is associated with a first state of the requested content, and wherein the first processing result corresponds to a first representation of the requested content for display on the first computing device; providing the first processing result to the first client computing device, wherein the first client computing device is configured to display the first representation of the requested content; obtaining a request to access the persistent browsing context from a second client-side browser application executing on a second client computing device separate from the first client computing device; and providing a second processing result to the second client computing device, wherein the second processing result is generated based, at least in part, on the first state of the requested content, wherein the first state of the requested content is based on one or more user interactions with the first representation of the requested content displayed at the first client computing device, wherein the second processing result corresponds to a second representation of the requested content for display at the second client computing device, and wherein the second client computing device is configured to display the second representation of the requested content.