My walking onion clumps don't die, but keep slowly expanding. The older clumps have shallot sized bulbs that I harvest as a way to rejuvenate the clump.
I now plant out only the biggest topsets to get new clumps, but when I first got them I planted all topsets. I have them all over the garden now. They only produce topsets once a year.
The flowers will sometimes produce seed for me. I've read it's not viable, but I keep dropping the ball on testing that out for myself.
In my climate, the clumps grow greens first thing in the spring and often again in the fall when it starts raining again.
To multiply the ones you have quickly, I'd just divide up the topsets and plant each individual bulbil. I usually plant the whole topset, cause I put them in inhospitable places and some always die.
Walking onions are cool cause the
roots from the bulbils will actually pull the bulb down into the ground. Since they do that, I never bury them when planting. I just make sure the base of the bulbil has good soil contact and let it decide how deep in the soil in wants to go. When I plant the whole topset, I make a little depression in the soil to put the topset in. That way the depression can hold a bit of moisture.
Welsh onions (A. fistulosum) are
perennial, so they'll keep going after flowering. I have to keep mine from being buried in deep snow or they sometimes die over the winter.