Final answer:
The territory between the Nueces River and the Rio Grande was controlled by Mexico until the 1870s, despite claims and disputes by the United States, which argued the Rio Grande as the border after annexing Texas.
Step-by-step explanation:
Between the Nueces River and the Rio Grande, the territory was controlled by Mexico until the 1870s. This was a matter of dispute, however, as the United States claimed the Rio Grande as the border after annexing Texas. Mexico, on the other hand, considered the Nueces River the legitimate border and hence believed it held sovereignty over that territory until the area was ceded to the United States following the Mexican-American War, which took place from 1846 to 1848.
During this period, there certainly were other influential forces at play in the region, for example, Native American tribes such as the Apaches and Comanches, who had their own claims and influences on the land. Nonetheless, in terms of official, recognized control, it was Mexico that had governance over the region until the post-war treaties came into effect, transferring control to the United States.