What happens with cards hosted on Builder after moving it to another server?
From FAQ:

Hosting
The following replaces the rules on hosting that appear in the
core set rulebook:
“Hosting” is the result of placing a card, counter,
or token on top of a card, creating a relationship between the host
card and what is hosted. If a card allows other cards to be hosted on
it, those cards must be hosted on the card when they are installed,
unless a card says otherwise.
Hosted counters or tokens can be spent
without affecting their host. If a trigger cost requires one or more
hosted counters or tokens, those counters or tokens are “spent” by
being returned to the token bank from the card the ability appears on.
If a host is trashed or uninstalled, all cards, counters, and tokens
hosted on it are also trashed. This cannot be prevented. If a host
Corp card is derezzed, all cards, counters, and tokens hosted on it
remain hosted.
The state of hosting is distinct (but not exclusive
from) the state of installing. Most cards are hosted on another card
when they are installed.

So, conditions under which hosting ends are &quot;trashed or uninstalled&quot;, moving doesn't affect hosting. Anything hosted on Builder will remain hosted on Builder after it moves.