Virtual water trade is widely considered as a potential method to solve local water shortage and unequal distribution. However, limited research investigated water-saving efficiency and water inequality of inter-provincial virtual water trade. In this study, we sought to explore this issue within China based on the 2015 input-output data. A multi-regional input-output model and a modified input-output model were used to estimate the virtual water trade and its impact on water-saving and water inequality. Our results suggest that: (1) The total virtual water flow across the country is 200.03 × 10⁹ m³, which accounts for 32.77% of water withdrawal. The agriculture sector contributes the highest proportion (73.99%) to virtual water flow. (2) Virtual water trade could decrease water withdrawal by 446.08 × 10⁹ m³ compared with withdrawal under no-trade situation at a national level, and 22 provinces could gain benefits through inter-provincial trade with a positive water-saving efficiency index. (3) Virtual water trade also causes water inequality, which exacerbates water scarcity of exported provinces, especially in northwest provinces. (4) There is a conflict between water conservation and water inequality, but different provinces show significant heterogeneity.

Shrinking cities are generally experiencing decreases in population, economic activity, and spatial expansion. However, whether this “low-growth” trajectory leads to an actual reduction in carbon emissions or is constrained by carbon lock-in effects and the complex interaction between urban shrinkage and carbon emissions remains unclear. To address this gap, this study examines 34 shrinking cities of the three northeastern provinces in China, utilizing nighttime light data to identify the spatial–temporal patterns of carbon emissions from a multidimensional perspective. Additionally, it explores the key drivers behind these emissions. Results show the following: (1) Spatiotemporally, carbon emissions are closely linked to shrinking cities, which also exhibit spatial–temporal heterogeneity. (2) There is a significant negative spatial correlation between carbon emissions and urban shrinkage degree (SD), with HL clusters (high–low clusters) and LH clusters (low–high clusters) being the main clustering types. (3) Through population, economic, and social driving factors, this paper identifies three synergistic effects shaping spatial–temporal carbon heterogeneity: passive reduction in economic scale (scale effect), volatility effect of structural transformation (structure effect), and spatial–institutional carbon lock-in (lock-in effect). The findings offer new insights into the low-carbon transition potential of shrinking cities and provide a basis for developing targeted policy frameworks to facilitate their sustainable transformation.