JY, as it relates to D3 to D1, the odds are stacked against the player for such a transfer, even though there is D1 talent at the D3 level.
Taking Rick's information, it is not likely, but is possible a D3 player with D1 talent could be classified as not recruited at the D3, by NCAA standards.
Assuming the D3 was a recruited athlete, here are the challenges to transferring:
1.) they need to follow the one year sit rule;
2.) they are not allowed to be offered or accept any athletic money during that one year;
3.) they probably lose significant units/credits which won't transfer, meaning they will be looking at, at least, a 5th year of college and tuition;
4.) the player risks everything and the college/coach risks not much of anything with such a transfer.
Before the one year sit rule, there was movement from D3 to D1 despite numbers 2-4. When we add the one year sit rule, the amount of movement has dwindled so it happens very, very seldom.