This paper studies the neural network-based distributed constrained $k$-winners-take-all ($k$WTA) problem, which aims to select $k$ largest inputs from amount of inputs under two types of global coupled constraints. Namely, equality and inequality constrained $k$WTA problems. By selecting the proper parameter, the two constrained $k$WTA problems can be transformed into two continuous constrained quadratic programming problems. Subsequently, we propose a derivative feedback-based modified primal-dual fully distributed algorithm for the $k$WTA problem with a global coupled equality constraint by utilizing Karush-Kuhn-Tucker (KKT) conditions and the gradient flow method. In addition, the developed derivative feedback-based distributed neurodynamic method is initialization-free. Furthermore, the above method is revised via a maximal projection operator for the $k$WTA problem with a global coupled inequality constraint. The two methods are rigorously proved to asymptotically solve the distributed constrained $k$WTA models in accordance with LaSalle's invariance principle. The performance of our designed methods is tested via four simulation examples.