Final answer:
The process where mentally ill individuals released from asylums end up in jails is known as the criminalization of mental illness, a trend stemming from the lack of adequate mental health care post-deinstitutionalization.
Step-by-step explanation:
The process wherein mentally ill individuals who are released from asylums end up in prisons or jails is known as criminalization of mental illness. After their release, due to the lack of proper mental health treatment facilities, support systems, or understanding of their conditions, many individuals with mental health issues find themselves in situations that lead to incarceration. This trend became noticeable particularly after the deinstitutionalization movement in the mid-20th century, when asylums and mental health hospitals were closed en masse, and such individuals were left with inadequate community-based services. Reformers like Dorothea Lynde Dix worked tirelessly to improve treatment for the mentally ill and to establish state mental asylums where they could receive proper care rather than punishment.