We will discuss a family of exclusion processes in one spatial dimension, where the random walk particles feel a drift whose speed depends on the local particle configuration. We show that their height fluctuations have a large-N limit given by the KPZ equation with an additional linear transport term. To our knowledge, it is the first KPZ result for a class of particle systems where the invariant measures are not explicit or known. This result also extends a prior series of works on deriving KPZ from stochastic Hamilton-Jacobi equations of Hairer, Quastel, Shen, Xu, and others.