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Development of the combustion process in the gaseous mixtures of near-limit composition is of great interest for fundamental aspects of combustion theory and fire-safety applications. The dynamics of ultra-lean gaseous flames in near-limit mixtures is governed by many effects, such as buoyancy, preferential diffusion, radiation, and instability development. Though ultra-lean combustion was extensively studied in microgravity conditions, the influence of gravity on the ultra-lean flame structure and stability is still poorly understood. The paper is devoted to deepening the knowledge of ultra-lean flame dynamics in hydrogen-air mixtures under terrestrial gravity conditions. The spatial structures of the flame developing under the effect of buoyancy forces are investigated employing detailed numerical analysis. Different modes of near-limit flame evolution are observed depending on the mixture concentration. In particular, we registered and described three distinct spatial structures: individual kernels tending to extinguish in leanest compounds, complex multi-kernel structures in marginal compositions, and stable cap-shaped flames in more chemically active mixtures. We apply the flame-bubble analogy to interpret flame dynamics. On this basis, the diagram in the Re-Fr plane is developed. That allows classifying the emerging flame structures and determine flame stability. Additionally, different ignition modes are studied, and the mechanisms determining the impact of ignition mode on the flammability limits are distinguished. Obtained results provide useful insights into the processes of flame quenching and development in near-limit hydrogen-air mixtures under real gravity conditions and can be applied in the design of contemporary fire-safety systems.