I have them all over the garden; have moved them; have taken "pups"
to plant elsewhere. They seem to do well --for a while. Then the
leaves get ragged-looking after the flower fades. Is this normal? Or
should the same plant bloom again later?

I will post a picture if need be, but meantime have found this site:

http://www.bromeliads.co.nz/lightshade.html

that has a plant much like mine. It is Aechmea Fasciata "Morgana".

Can anybody comment on whether I should pull up the ragged-looking
ones -- or will they bloom again?

Thanks.

Persephone