Firefox 19 and later now support these shortcuts natively:
Firefox 19 added shortcuts to move tabs:
- Ctrl+Shift+PgUp to move left, and
- Ctrl+Shift+PgDn to move right.
The change is recorded as bug #364845; here is the specific commit that added the shortcuts.
Therefore, the Move Tabs extension is no longer required; in fact, if you have it installed you should disable it (or else the shortcut will move the tab twice).
For moving tabs to a new window, the Tab Mix Plus extension provides the shortcut Ctrl+Alt+N (this can be customized in the extension's options). For moving the tab into a different window, there's no default shortcut, but you can set it up by changing the merge behavior (Tab Mix Plus Options > Events > Tab Merging) to merge only the current and last-focused windows, and then setting a shortcut for Merge (the default shortcut Ctrl-Shift-M conflicts with a built-in one).