How does one apply Kalman smoothing with irregular time steps?

I would like to apply Kalman smoothing to a series of data sampled at irregular time points. There is a claim on Stack Exchange that “For irregular spaced time series it’s easy to construct a Kalman filter”, but I haven’t been able to find any literature that specifically addresses this.

In my situation, I’d like to use a simple exponential covariance relationship to reflect the idea that the underlying continuous process is evolving as a linear dynamical system from which we irregularly receive samples.

So: is it simply OK to apply a Kalman filter with the “predict” step using a transition model and a process noise model whose “amplitude” depend on the amount of time that has elapsed since the last measurement?

Yes. In fact, this is how the Kalman Filter (KF) is also set up, at least implicitly. The assumptions in place when choosing the KF model, are that the movements and measurements compose a linear dynamical system. The transition matrix, $F_{t-1}$, (in the equation: $\hat{x}_{t|t-1} = F_tx_{t-1} + …$, where $\hat{x}$ is the predicted state estimate) is in fact indexed by time, so irregular observations shouldn’t be an issue.