Consider Builder with installment(s). A Patch for example. What happens with it after the Builder being moved to another server? Nothing, I suppose? Seems like all the installments move with the Builder.
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 "trashed or uninstalled", moving doesn't affect hosting. Anything hosted on Builder will remain hosted on Builder after it moves.