Parsing texts into universal dependencies (UD) in realistic scenarios requires infrastructure for the morphological analysis and disambiguation (MA&D) of typologically different languages as a first tier. MA&D is particularly challenging in morphologically rich languages (MRLs), where the ambiguous space-delimited tokens ought to be disambiguated with respect to their constituent morphemes, each morpheme carrying its own tag and a rich set features. Here we present a novel, language-agnostic, framework for MA&D, based on a transition system with two variants — word-based and morpheme-based — and a dedicated transition to mitigate the biases of variable-length morpheme sequences. Our experiments on a Modern Hebrew case study show state of the art results, and we show that the morpheme-based MD consistently outperforms our word-based variant. We further illustrate the utility and multilingual coverage of our framework by morphologically analyzing and disambiguating the large set of languages in the UD treebanks.