You can move them into a nursery bed at any time, provided you do it carefully.

The decline in the main bulb(s) - provided nothing else in the bed has grown up to shade them out - is probably due to overcrowding. Best to wait until they are just going over to lift and divide the clump. In the meantime make sure to feed them well this year.

Thanks Kayman, I do not have anything shading them so the decline must be due to crowding. What do you mean by "going over" before I lift and divide?

Leave them until the foliage starts to yellow, a few weeks after they have flowered. Replant at the depth of the lowest bulbs in the clump.

Another thing that could be making your lilies smaller is cutting too much. If you cut more than 1/3 to 1/2 the plant, it can't get enough energy from its remaining leaves to be bigger the next year. There really are four things that can make the plant shrink from one year to the next: not enough food, overcrowding, shade, and cutting too much.

You can actually dig and replant any time, if you are careful, but kayman's advice will give you the biggest plants for next year. Anytime you dig, you are interrupting the flow of water and nutrients into the plant, until the roots re-establish, so it's best to dig when the leaves and stems turns yellow. Lilies never go completely dormant - their roots are always active - so DON'T break off the roots at the base of the bulbs, keep them out of the ground as briefly as you can, and spread the roots out as well as you can when you replant. If you're replanting any really small bulblets, plant them more shallowly; about twice as deep as the bulb is tall is about right. Those itty-bitty bulblets won't bloom for a year or two, so you might want to plant them in an inconspicuous place.

Jeanne