First, a landlord cannot just expel a tenant, this involved getting a court order where the judge tells the tenant to leave. Second, she might think that "You're not a tenant, you're just living here" but that is pretty much what it means to be a tenant – she agreed to let the relative live there. There does not have to be a detailed lease written up, though the term of such unwritten agreements is usually month-to-month (this is all about the laws of the particular state, but no state authorizes instant expulsion).
You would need to look at the laws of your state for the "abuse" question. Using Washington as an example, and trying to make the best case possible that bailing out is against the law, the potential legal concern is "abandonment", which is defined as an
action or inaction by a person or entity with a duty of care for a
vulnerable adult that leaves the vulnerable person without the means
or ability to obtain necessary food, clothing, shelter, or health
the failure of a vulnerable adult, not living in a facility, to provide for himself or herself the goods and services necessary for the vulnerable adult's physical or mental health, and the absence of which impairs or threatens the vulnerable adult's well-being. This definition may include a vulnerable adult who is receiving services through home health, hospice, or a home care agency, or an individual provider when the neglect is not a result of inaction by that agency or individual provider.
RCW 74.34.200 creates a legal cause of action for abandonment, abuse, financial exploitation, or neglect of a vulnerable adult. Then one might sue for abandonment if one is living at home and
receives care from a home health, hospice, or home care agency, or an
But, "individual provider" is a defined legal term:
a person under contract with the department to provide services in the
home under chapter 74.09 or 74.39A RCW
If the tenant is under contract with the state to provide services, then abandoning the vulnerable adult could get you sued. The courts do not declare that a duty of care exists because of a genetic relationship. A duty of care might be found from a contractual relation (one not involving a state social services agency), for example "You can live here if you take care of me". So there is a legal concern over abandonment, which the person should talk out with an attorney (which is possible since instant-eviction is not an option). Said attorney might also discuss the concept of guardianship.