Digging and replanting tulips is essentially what the tulip grower does, so in theory you could grow any cultivar if you do so, as long as you have no trouble with diseases. Of course a light very well drained soil will still be very beneficial! The ideal time to dig them would be as soon as the outer skin turns brown. Then they need a dry summer rest. Digging them at a later time would be much less ideal.
In practice you will probably find that many cultivars are prone to actually dividing to much and you will end up with a lot of small non flowering bulbs that need growing on further. This supposedly can be somewhat helped with deeper planting as tulips are less prone to divide then, versus being more stressed from shallow planting, but still there can be huge differences between cultivars.
Because of deer, since a few years I only grow tulips in protected sites, in pots, for cutting in the kitchen garden and I also grow them on to get large enough bulbs for the pots. Every year I throw away hundreds of small bulbs. Most tulips will multiply like you wouldn't believe when you do it this way.
