The name of your uni is not going to matter to admissions tutors but the curriculum is.
If you want to do an MSc in Economics at a top uni for instance, the odds are the MSc will have a lot of advanced maths in it, so the tutor will want to know you have done enough at undergrad to be able to pick it up at the start as the MSc moves too fast to do remedial work. If you went to a mid or lower ranked uni and did a BA Economics option which basically allowed you to choose the modules you wanted and you've done about A-level maths and no higher, then you wouldn't have done the necessary background knowledge to do their MSc so they won't take you. But if say you did BSc Economics with Statistics and you have a module list that includes quite a lot of maths then the admissions tutor will probably ask you what textbooks you used, what topics you've studied and if it sounds like you've done what they need then they will be ok with it, they aren't going to say "oh your past university isn't as good as ours therefore you aren't good enough for us".
It's kind of like football transfers. When Luis Suarez was joining Liverpool, Liverpool didn't say "he plays in the Dutch league which is second rate so he's not going to be good enough for us" they went ok, if he comes to the Premier League he's going to have to be able to do x, y and z to be a force, so lets watch his games and do our due diligence on if he has those skills. Same with Henrik Larsson going from the Scottish league to Barcelona.
The big constraint in terms of studying a Masters at a top uni that often students pre-undergrad don't understand, is the level of fees, you are often talking £15,000 to £25,000. If you think "well surely there must be scholarships that fund that" you may be in for a nasty surprise.