Would it be beneficial for me to transfer to a university with a PhD program instead of continuing to study for a MS?
I can only speak from my experience. To note, I am a recently graduated PhD from a top school in an engineering field. I served on an application committee as a student member. Two main issues come to mind:
1) How will your decision impact your possible acceptance into a PhD program, and
2) How will your decision influence your finances (course fees, possibility of courses taken that will not count towards your degree.)

The application process for PhD programs starting fall 2016 have closed, to the best of my knowledge. I do not know the likelihood of transferring within a Masters program - even then, you would still need to submit a PhD program application at the same school when it is next open, though you would be better set assuming you have an advisor and research ongoing.

Even with a Masters at the same school, same department, classwork does not necessarily count towards your PhD requirements.  Be sure to check if it does. 

If you decide to finish your Masters at your current school, your future PhD application will be viewed a little differently from a straight from undergrad application. As someone who has completed their masters and who has dedicated time specifically to research, you will be expected to have at least one publication in a refereed top conference/journal. An undergraduate should have evidence in their work/project history that they are capable of doing research, but it is not critical they have a publication.  This may vary depending upon the school and department, but keep it in mind.  

There are sometimes Master degrees that follow one of two plans (coursework Masters or project/research Masters). A coursework Masters is at a significant disadvantage compared to other Masters when doing PhD applications.

I don't have complete knowledge of your situation, or goals, of course, but if I was in your situation, I would try to do the following: pick and contact a couple schools to check out the possibility of a transfer within the Masters program for this term. If yes, establish contact with a couple of professors about research in their lab. With a masters at the school you plan to apply to for PhD, you can immediately start your research, develop connections, have a better chance of getting accepted into that PhD program, and are more likely to have coursework count towards your PhD requirements.  If the transfer is not possible, I would complete the Masters at the current school while emphasizing research, pushing for a paper which is good evidence that capable of doing solid work.