<p>I’m not in engineering, so perhaps its different in that field, but in almost every other field, if you have a masters or PhD from a top school, no one cares where you did your undergraduate degree. </p>
<p>Graduating at the top of a less competitive school with outstanding letters of recommendation from faculty who know you well and who have provided you with good research opportunities would be much better than transferring to and graduating from a better known school but with faculty who know you less well and therefore provide less strong letters of rec and give you few research opportunities. (And if the school is more competitive, you may not stand out as much academically). </p>
<p>So, I think the real question is whether you are happy at your current school: Do you like the people, the culture, the faculty, the work/play balance? If the answer is yes, then I would stay put if it were me.</p>