Transplanting when sprouting begins....
If you mean when you first see the root (not the leaf) emerging from the seed, then yes, that's a good time to transplant. The seed will continue to concentrate on root growth and repair, before extending the leaf.
But if you are waiting for the leaf to emerge, that is a very stressful time, and I think this is your first problem. At this early stage, lilies just are not so resilient to root damage, like a tomato (or daylily) seedling, for instance. No matter how careful you are, you can't transplant without massive damage to such a tiny delicate lily root. Now the seed must try to maintain and repair the leaf and the root with very little reserve energy to draw from. At this time, it depends on the ability that it should have gained by that time to absorb nutrients through the roots and make energy from sunlight in the leaf to sustain itself. Alas, this is heavily curtailed, due to your intervention.
I am assuming you have some kind of a hood to keep humidity high after transplanting, so this would account for the deaths happening in the ensuing first week or perhaps a little longer. I would expect the soil moisture to be the problem as Steve mentions for later fatalities - likewise for any seeds that produce a bulb first and a leaf the next year (called delayed hypogeal germination -DH).
Martagons and most orientals are DH. Asiatics emerge the first season of germination. But sometimes, asiatics do wait a year, and complete the entire germination and sprouting process in the second year.