What an interesting question. Since border patrol is tasked with monitoring the inflow of people to the United States, their job is more akin to law enforcement than to immigration services. However, with a large percentage of individuals at the U.S. southern border trying to immigrate, the conflation of duties for border patrol results in muddied priorities.
A functionalist would approach crimmigration at the border as a necessary combination of policies and procedures. Because the number of personnel needed to process various types of immigrants—lawyers, judges, social workers, translators—is not high enough to meet the need, border patrol agents have been asked to oversee the filtering of potential immigrants into the U.S. This has resulted in detention, a domain of law enforcement, instead of bureaucracy, a domain of immigration. Though functionalists might analyze the commingling of these roles, in general they would consider the solution as one designed to meet an immediate need.
A conflict theorist would approach crimmigration at the border by determining the power dynamic, who benefits from the dynamic, and who suffers because of it. Because law enforcement tactics are being used to manage the inflow of individuals at the border, it is easy to discern that border agents, as representatives of the U.S. government, hold the power in the relationship. It seems clear, then, that those who suffer as part of the equation are the individuals trying to cross into the United States. However, the question of who benefits from the dynamic would require further analysis. Some potential beneficiaries, according to conflict theorists, could be politicians, media outlets, and government contractors, to name a few.