Some problems with bikes that old:

The original 5-speed rear dropout spacing was 120 mm. When standard 6 and 7-speeds were developed the spacing was increased to 126 mm and for 8-speed and up road bikes went to 130 mm and MTB's to 135 mm. Your old frames are likely 120 mm and will have to be "cold set" (aka bent) to accept wider hubs with out a struggle. Typically cold setting a 126 mm frame to 130 is considered easy, 120 to 130 not so much.

Your Raleigh had proprietary bottom bracket and fork steerer threading and won't accept the far more common "English" threaded bottom brackets and headsets. There are modern bottom bracket cups made to fit that threading by Phil Wood but they are very expensive.

I'm not sure about your Schwinn but if it has an Ashtabula crank, there are adapters that will convert it to take an modern bottom bracket. BTW, here are a couple of articles on the history of those early Varsities: http://sheldonbrown.com/varsity.html
http://www.cyclesportandfitness.com/the_varsity.htm

Finally, your Raleigh probably had 27" wheels and modern 700c wheels may not align with your current brakes if the pad adjustment range (4 mm lower) isn't sufficient.

I'm sure there are other mis-matches i haven't mentioned but these are the ones that came to mind.