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I like the idea that pieces might change sides during the game.  This
effect could be associated with the temporal jumps.  Obviously, a change
in the space-time continuum might result in such shifts.

What would be the condition of such shifts?

I would restrict temporal jumps specificially to the Doctor and the Master
piece.  But they each might be able to take associate pieces with them. 
The temporal jump could be seen as a shock-wave which results in any
pieces which are threatened or defended by this piece(and its
companion[s]) shifting sides.  Gaining an opposing piece would definitely
be positive.  And possibly losing a friendly piece might be the price to
pay for such a move.

Temporal jumps would need be controlled.  There needs to be very
restrictive conditions for such.  Ones that allow the opposing player the
opportunity to deny a player this ability.  This might be associated with
the control of a key and/or a particular piece.  This actually might be a
Tardis piece which is non-moving, except when it space-time leaps.  The
control of this piece might be based upon possession of its key.