It would depend on the moisture content as to whether the spalting stops or not within the spalted piece, but normal air dry is usually sufficient to halt that anyway, as for it spreading to other pieces, that is not possible unless the fungus causing the spalt develops fruit bodies (usually in the form of a bracket fungus but not always, some toadstool shaped fungi can also be a part of the process as well as various club shaped ones)
But with all of these it is the fruit body that carries the spores necessary for the fungus to move to a new piece of wood, on top of that the conditions have to be just right too, think of a woodland floor in mid to late autumn, wet, cool and shady, those are the required conditions for the fungus to take hold. What we see in the patterns of Spalting is the action of the main part of any fungus, the mycellium, which can be basically considered as the root, but in fact is the true form of the fungus, the white areas of spalted wood are caused by a family of bracket fungus, turkey tail being a prime one.
The black lines are demarcation lines between different mycelliums, they mark a border and somehow prevent other fungi from encroaching.
So, store your spalted wood alongside any other timber in the dry with confidence, there is no risk of it moving across.