To insure success with them, wait until the stems have yellowed/browned out, become ratty and woody and any foliage is yellow and droopy - that means the bulb has taken the energy it needs to replenish itself for next season's growth. For ease when digging up, cut the stem down to 5-6 inches above the soil line; once out of the ground, the rest of the woody stem can be cut away a bit above the bulb. Transplant them where wanted without waiting too long. Here in zone 4, I don't water them - rain will be sufficient.

I have lots of these - because I think they're charming and because just about every one of the bulbils that drop naturally (or with help) sprouts.

Your second picture is a daylily.

And the little bulbils between the stem and leaf axils are seeds. Those can be removed now and replanted immediately just barely under the soil where you want them. It'll take a few growing seasons - 3 to 4 seasons before they reach blooming size.

Can't advise much on the propagation aside from eventually digging up any clump and separating the bulbs. That one looks to be growing from only one bulb at this point. Each bulb throws up one stem - multiple stems will be your guide.

Aside from eventual bulb separation - and if this type doesn't produce the little bulbils between the stem and leaf axils like the tigers - one of the real experts here would have to weigh in on any other propagation methods.