Derivations under different grammar formalisms allow extraction of various dependency structures. Particularly, bilexical deep dependency structures beyond surface tree representation can be derived from linguistic analysis grounded by CCG , LFG , and HPSG . Traditionally, these dependency structures are obtained as a by-product of grammar-guided parsers. In this article, we study the alternative data-driven, transition-based approach, which has achieved great success for tree parsing, to build general dependency graphs. We integrate existing tree parsing techniques and present two new transition systems that can generate arbitrary directed graphs in an incremental manner. Statistical parsers that are competitive in both accuracy and efficiency can be built upon these transition systems. Furthermore, the heterogeneous design of transition systems yields diversity of the corresponding parsing models and thus greatly benefits parser ensemble. Concerning the disambiguation problem, we introduce two new techniques, namely, transition combination and tree approximation, to improve parsing quality. Transition combination makes every action performed by a parser significantly change configurations. Therefore, more distinct features can be extracted for statistical disambiguation. With the same goal of extracting informative features, tree approximation induces tree backbones from dependency graphs and re-uses tree parsing techniques to produce tree-related features. We conduct experiments on CCG -grounded functor–argument analysis, LFG -grounded grammatical relation analysis, and HPSG -grounded semantic dependency analysis for English and Chinese. Experiments demonstrate that data-driven models with appropriate transition systems can produce high-quality deep dependency analysis, comparable to more complex grammar-driven models. Experiments also indicate the effectiveness of the heterogeneous design of transition systems for parser ensemble, transition combination, as well as tree approximation for statistical disambiguation.