Some few seedlings it is almost impossible to transplant, can be done but difficult. At this moment examples escape me.

Root crops are very difficult to transplant.

Others will look sickly for a half day or so. Misting and protection from light will help. Transplanting in the evening is often a good idea.

You say tiny, how tiny?

They are mostly just bearing the first leaves; that's how tiny they are. Some seedlings are not as big as sesame seeds.

Umm, most transplants should have at least two "real" leaves, otherwise their newly forming baby roots are too tender and liable to break. If you are desperate, sometimes you can use a spoon to lift a whole chunk of soil so that it still encases and protects the roots.

It's usually better to space the seeds when starting, so that they can at least grow two leaves before bumping into their neighbor. Spacing is very hard to do with fine seed, and one usually mixes the seed with powdered gelatin to force spacing. Else you snip the too-close neighbors to give growing room around.