I sometimes want to move question to another community manually instead of flagging it. Can I do it and if I can, how many reputation needed for that action?
When you reach 3000 reputation you can cast close votes on questions.
One of the close reasons is "Off Topic" and when choosing it, you're given list of some relevant sites that you can choose from.
For example on Stack Overflow you get this dialog:
When a questions have 5 close votes, it's automatically Closed.
If most close votes (4 or more) are "off topic" and chose the same site, the question is then automatically migrated to that site. Anything fewer and the question is just closed as "off topic".
If the person who posted the question has no account in the target site, the question author will appear gray there until he log in and associate the accounts.
Note: if the question author has account on the target site, and that account is blocked from asking questions (or is under suspension) the question can't be migrated, even by a moderator. It will still be closed, but even 100 votes won't migrate it.