The possibilities for foreign students from third countries with higher education qualifications acquired in Hungary are described in details in the linked document: 1) seeking a job or starting a business in Hungary 2) Hungarian Card 3) EU Blue Card 4) guest self-employment 5) Research 6) National Card
As for the status of visiting lecturers, language consultants and researchers, see and scroll down to the description below: 'Conditions of Residence in Hungary for Research Purposes'

Major changes that are still in effect since 1st January, 2024

An outline:

• An extension procedure from Hungary cannot be started if your residence permit expired. Not even with an equity request. The applicant has to go home and start a new visa case.
• A visa to enter Hungary (D or C) is valid only for 3 months after decision-making (not from the date of the receipt)!
• Other purpose ceased to exist! Instead: research / job-seeking / work
• Family unification with study purpose res.card is only permitted for those who possess a residence permit issued before 1st January, 2024. Exception: a baby born in Hungary.