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Terrestrial ecosystems currently absorb a substantial proportion of anthropogenic carbon dioxide (CO₂) emissions, yet the persistence of this land carbon sink under accelerating climate change remains largely uncertain. While process-based vegetation models project continued enhancement of carbon uptake through CO₂ fertilization, there are growing observational evidence that suggests that climate extremes, particularly 3oC global warming level, may substantially constrain or reverse these gains through droughts and heatwaves. Recent global carbon budget assessments also show a rapid weakening of the land carbon sink, with dynamic global vegetation model ensembles and atmospheric inversions indicating a large decline in net land uptake between 2022 and 2023. Here we present an observation-constrained, species-specific quantification of global terrestrial carbon fluxes across major terrestrial biomes. We integrate a combination of satellite-derived vegetation indices, a geo-referenced global database of climate-induced tree mortality, and dynamic global vegetation models from the TRENDY intercomparison models to map spatial and temporal variability in carbon uptake, storage, and loss under historical and future climate conditions. Model simulations are forced with regionally downscaled climate projections and explicitly constrained using observed mortality signals to quantify the effects of CO₂ fertilization. Our results reveal a widespread divergence between modelled and observation-constrained carbon fluxes, with some biomes exhibiting a reduced carbon sink. Species adapted to moderately moist climate conditions show strong sink-to-source transitions, while drought-tolerant species exhibit greater resilience but limited long-term sequestration capacity. These findings demonstrate that climate extremes impose substantial limits on terrestrial carbon sequestration. By linking species-level ecological responses with carbon flux dynamics, our study provides a more realistic assessment of the future role of terrestrial ecosystems in regulating the global carbon cycle under ongoing climate change, as well as show species and regions for optimal sequestration.Key words: Carbon Flux, Carbon Dioxide Removal, Biomes, Dynamic Vegetation Models, Drought, Heatwave