Land carbon sinks are responsible for removing about a quarter of anthropogenic CO₂ emissions, and make up approximately half of total global carbon sinks. Uncertainty in the response of land carbon sinks to climate and changing atmospheric CO₂ are large, and dominate the uncertainty in total carbon sinks under future climate. Understanding the carbon cycle response to net-zero and net-negative emissions has important implications for projecting future climate. Experiments in the "flat10" model intercomparison were designed for directly estimating key climate metrics that underlie carbon budgeting frameworks. Here we characterize the response of land carbon pools and fluxes from ten emissions-driven Earth system models (ESMs) under positive, net-zero, and net-negative CO₂ emissions. Although there are many differences in simulated land carbon pools and fluxes across models, we find some consistent behavior across ESMs. 1) During the positive emissions phase, carbon is gained on land primarily in vegetation pools. 2) Following net-negative emissions to the point of cumulative zero emissions, carbon is lost from land in tropical latitudes, primarily from vegetation pools, but in mid- and high-latitudes most models show net land carbon gain, primarily in soil pools. 3) Following an extended period of net-zero emissions, a majority of models again show carbon gain in mid- and high-latitudes and vegetation carbon loss in the tropics. Under net-negative emissions the timing of vegetation carbon response relative to peak emissions is relatively consistent across ESMs, but timing of soil carbon response varies widely, implying larger intermodel disagreement associated with responses of soil carbon which tends to have longer timescales relative to vegetation carbon. Our findings highlight that tropical carbon is most likely to be both gained and subsequently lost under positive, zero, declining, and negative emissions, with possible implications for carbon dioxide removal efforts.