We establish asymptotic expansions for nonautonomous gradient flows of the form (u) over dot(t) = -del f(u(t), r(t)), where f(x, r) is a penalty approximation of a linear program and the penalty parameter r(t) tends to 0 as t -> infinity. Under appropriate conditions we show that every integral curve satisfies u(t) = u(infinity) + r(t) d(0)* + (r) over dot(t)r(t) w(0)* + o((r) over dot(t)r(t)) for suitable vectors u(infinity), d(0)*, and w(0)*. We deduce an asymptotic expansion for a related dual trajectory, and we show that the primal-dual limit point is a pair of strictly complementary optimal solutions for the linear program.