  In inverse problems, the goal is to estimate unknown model parameters from
noisy observational data. Traditionally, inverse problems are solved under the
assumption of a fixed forward operator describing the observation model. In
this article, we consider the extension of this approach to situations where we
have a dynamic forward model, motivated by applications in scientific
computation and engineering. We specifically consider this extension for a
derivative-free optimizer, the ensemble Kalman inversion (EKI). We introduce
and justify a new methodology called dynamic-EKI, which is a particle-based
method with a changing forward operator. We analyze our new method, presenting
results related to the control of our particle system through its covariance
structure. This analysis includes moment bounds and an ensemble collapse, which
are essential for demonstrating a convergence result. We establish convergence
in expectation and validate our theoretical findings through experiments with
dynamic-EKI applied to a 2D Darcy flow partial differential equation.
