H1 transfer probable but lower salary offered by new potential employer

My H1 got approved - I am working with comp A for ‘x’ salary and ‘y’ job title.

Interviewed with a better Employer B who have indicated their willingness to transfer H1 - however the job is not the same - i.e. could be same broad job family but not the same title. Basically, the stipulated minimum wage for both jobs is different. I also came to know that the salary from Employer B may be lower than my current job.

Can a transfer be done in such a case? Employer B offers better long term growth and satisfaction - please let me know if transfer is even a possibility in such a situation?

Thank you!