The highly destructive Asian brown marmorated stink bug (Halyomorpha halys, BMSB) invaded Europe, Caucasus region, and North and South America. Efforts to control it are ongoing in the Palearctic European-Mediterranean Basin and North America by introducing and redistributing two Asian stenophagous scelionid egg parasitoids (Trissolcus japonicus and T. mitsukurii) that are attacked by an adventive oligophagous pteromalid Asian hyperparasitoid (Acroclisoides sinicus). Large BMSB nymphs and adults may be parasitized by new associations of oligophagous tachinid flies and immature stages by egg parasitoids and predators. The terms stenophagous and oligophagous are commonly used to define narrow and wider ranges, respectively, of food eaten, but here they refer to the range of hosts attacked by adult female parasitoids. A holistic weather-driven physiologically based demographic model (PBDM) of the tritrophic interactions was developed to evaluate prospectively the impact of natural enemies on the biological control of BMSB under current and climate change weather. Our study focuses on the European-Mediterranean region, with the results for the USA, Mexico, and Central America reported as supplementary information. The PBDM analysis suggests that biotypes of the egg parasitoids T. japonicus and T. mitsukurii with high search capacity could suppress BMSB regionally, but the requisite levels of parasitism by these parasitoids for economic control are not observed in their native range nor in invaded areas. The model suggests that the action of T. japonicus is greater than that of T. mitsukurii, but that joint interactions of the two egg parasitoids would provide higher mortality of BMSB. Field data and model results suggest that the egg hyperparasitoid A. sinicus has a modest negative impact on the suppression of BMSB. Moreover, tachinid parasitoids of adults could have an important supplemental role in suppressing BMSB densities. Analysis suggests that new biotypes of egg parasitoids and species of tachinid parasitoids of large nymphs and adults be sought.