Multimodal interfaces are becoming increasingly ubiquitous with the advent of mobile devices, accessibility considerations, and novel software technologies that combine diverse interaction media. In addition to improving access and delivery capabilities, such interfaces enable flexible and personalized dialogs with websites, much like a conversation between humans. In this paper, we present a software framework for multimodal web interaction management that supports mixed-initiative dialogs between users and websites. A mixed-initiative dialog is one where the user and the website take turns changing the flow of interaction. The framework supports the functional specification and realization of such dialogs using staging transformations -- a theory for representing and reasoning about dialogs based on partial input. It supports multiple interaction interfaces, and offers sessioning, caching, and co-ordination functions through the use of an interaction manager. Two case studies are presented to illustrate the promise of this approach.