Recently, Mobile-Edge Computing (MEC) has arisen as an emerging paradigm that extends cloud-computing capabilities to the edge of the Radio Access Network (RAN) by deploying MEC servers right at the Base Stations (BSs). In this paper, we envision a collaborative joint caching and processing strategy for on-demand video streaming in MEC networks. Our design aims at enhancing the widely used Adaptive BitRate (ABR) streaming technology, where multiple bitrate versions of a video can be delivered so as to adapt to the heterogeneity of user capabilities and the varying of network condition. The proposed strategy faces two main challenges: (i) not only the videos but their appropriate bitrate versions have to be effectively selected to store in the caches, and (ii) the transcoding relationships among different versions need to be taken into account to effectively utilize the processing capacity at the MEC servers. To this end, we formulate the collaborative joint caching and processing problem as an Integer Linear Program (ILP) that minimizes the backhaul network cost, subject to the cache storage and processing capacity constraints. Due to the NP-completeness of the problem and the impractical overheads of the existing offline approaches, we propose a novel online algorithm that makes cache placement and video scheduling decisions upon the arrival of each new request. Extensive simulations results demonstrate the significant performance improvement of the proposed strategy over traditional approaches in terms of cache hit ratio increase, backhaul traffic and initial access delay reduction.