The availability of climate model experiments under three alternative scenarios stabilizing at warming targets inspired by the COP21 agreements (a 1.5 °C not exceed, a 1.5 °C with overshoot and a 2.0 °C) makes it possible to assess future expected changes in global yields for two staple crops, wheat and maize. In this study an empirical model of the relation between crop yield anomalies and temperature and precipitation changes, with or without the inclusion of CO2 fertilization effects, is used to produce ensembles of time series of yield outcomes on a yearly basis over the course of the 21st century, for each scenario. The 21st century is divided into 10 year windows starting from 2020, within which the statistical significance and the magnitude of the differences in yield changes between pairs of scenarios are assessed, thus evaluating if and when benefits of mitigations appear, and how substantial they are. Additionally, a metric of extreme heat tailored to the individual crops (number of days during the growing season above a crop-specific threshold) is used to measure exposure to harmful temperatures under the different scenarios. If CO2 effects are not included, statistically significant differences in yields of both crops appear as early as the 2030s but the magnitude of the differences remains below 3% of the historical baseline in all cases until the second part of the century. In the later decades of the 21st century, differences remain small and eventually stop being statistically significant between the two scenarios stabilizing at 1.5 °C, while differences between these two lower scenarios and the 2.0 °C scenario grow to about 4%. The inclusion of CO2 effects erases all significant benefits of mitigation for wheat, while the significance of differences is maintained for maize yields between the higher and the two lower scenarios, albeit with smaller benefits in magnitude. Changes in extremes are significant within each of the scenarios but the differences between any pair of them, even by the end of the century are only on the order of a few days per growing season, and these small changes appear limited to a few localized areas of the growing regions. These results seem to suggest that for globally averaged yields of these two grains the lower targets put forward by the Paris agreement does not change substantially the expected impacts on yields that are caused by warming temperatures under the pre-existing 2.0 °C target.