I'm completing a PhD at a former polytechnic. There is an issue that makes me doubt my choice. I did very well for my Masters degree. I went to the ex-poly because I thought the supervisors would be ideal. However, I am beginning to realise they aren't really helping my cause.
If I finish the doctorate here, will I only be accepted at universities that have little funding where academics have to devote all their time to teaching? Are there many cases of people completing a doctorate at a small university and then finding their way to a internationally known, well-funded university? This is a question that haunts me and I don't feel I can approach my supervisors about it.
Any help would be appreciated.

Hi. I formed a project with a supervisor in order to get funding. However, funding went to someone who performed better in interview. Because I liked the project, though, I applied for funding from elsewhere and received a full grant. However, I'm not sure if I made the right choice as I perhaps could've used that funding at a higher-ranked institution with more of a research culture. I'm at an ex-poly (top 50).
I was keen to get onto the project, though, and really liked the supervisor's previous work. They had written extensively about my very niche topic and they know all the top names in the field. They are also doing research now that is related to this area and want to include me in it.
However, I can't get over the feeling I could be at a more prestigious place - I received top marks for BA and MA. A contact said she 'wouldn't do her PhD there'. I do like the supervisor and we get on well, though, and I like the person who got the initial studentship. Should I see if I can transfer to a bigger institution using my grant (perhaps there are one or two people who could supervise) or just carry on and make the most of my funding here? I can't seem to get over the fact I (possibly) could've used grant money for pretty much anywhere, depending on if I got a PhD place.

Thanks