We consider multi-component matching systems in heavy traffic consisting of $K\geq 2$ distinct perishable components which arrive randomly over time at high speed at the assemble-to-order station, and they wait in their respective queues according to their categories until matched or their ``patience" runs out. An instantaneous match occurs if all categories are available, and the matched components leave immediately thereafter. For a sequence of such systems parameterized by $n$, we establish an explicit definition for the matching completion process, and when all the arrival rates tend to infinity in concert as $n\to\infty$, we obtain a heavy traffic limit of the appropriately scaled queue lengths under mild assumptions, which is characterized by a coupled stochastic integral equation with a scalar-valued non-linear term. We demonstrate some crucial properties for certain coupled equations and exhibit numerical case studies. Moreover, we establish an asymptotic Little's law, which reveals the asymptotic relationship between the queue length and its virtual waiting time. Motivated by the cost structure of blood bank drives, we formulate an infinite-horizon discounted cost functional and show that the expected value of this cost functional for the nth system converges to that of the heavy traffic limiting process as n tends to infinity.