We study a class of two-player zero-sum Colonel Blotto games in which, after allocating soldiers across battlefields, players engage in (possibly distinct) normal-form games on each battlefield. Per-battlefield payoffs are parameterized by the soldier allocations. This generalizes the classical Blotto setting, where outcomes depend only on relative soldier allocations. We consider both discrete and continuous allocation models and examine two types of aggregate objectives: linear aggregation and worst-case battlefield value. For each setting, we analyze the existence and computability of Nash equilibrium. The general problem is not convex-concave, which limits the applicability of standard convex optimization techniques. However, we show that in several settings it is possible to reformulate the strategy space in a way where convex-concave structure is recovered. We evaluate the proposed methods on synthetic and real-world instances inspired by security applications, suggesting that our approaches scale well in practice.