hi, I have transplanted carrots sucessfully. but you need to transplant an awful lot to get anywhere $$$.
figure a 72 count flat if 80 % sucess rate is about 60 good carrots---8 to a bundle/7 bundles. figure 1.50-2.00 a bundle.
thats a lot of time/labor for $$10-12 bucks.
you would need to plant 20 flats for $200-$250 (1200 carrots-or 250 row ft.)

ive never tried to transplant beets, but they are very easy to sow. you can plant in moist soil as long as its not mud.
radish plant 1 seed every inch for no thinning.
do you have a row planter?

turnips -kohlrabi--same as beets
need good fertile soil-not hot weather (60-80 degrees best)
consistent water ( sidedress once root starts to form--10-30-10)

root crops should be thinned for large hi-quality vegetables
beets-3-4inch carrots 3 inch radish 11/2 inch kohlrabi
4 inch