Let me see if I understand. You have two outcome variables, call them O1 and O2, and you are interested in change between O1 and O2 (so O1 -> O2) and you think there is a mediator, call M, so O1 -> M -> O2 and you think the mediator association is related to some group variable G. It would be useful to provide the equations that you want to compare (or the path diagrams). If you just have two points you probably don't need to read much about longitudinal methods since most of the discussion for those have 3 or more measurements. You should probably consider the issues related to Lord's paradox (see references in www.degruyter.com/document/doi/10.1515/edu-2020-0107/html).

Hi Lindsey

the question touches two issue that I answer separately

a) Mediation: In a two-wave panel model, you can have either lagged effects or synchronous/contemperous effect. Lagged effects mean that X_t1 affects Y_t2. The will emerge when the time interval will be close to the causal lag. synchronous effects have X_t2 affect Y_t2; the will emerge if the causal lag is much shorter. With a mediation model you have X, M and Y measured at both waves. Hence, a lagged mediation effect would be estimated by X_t1 --> M_t2 and M_t1 --> Y_t2 and a synchronous mediational effect would be tested by X_t2 --> M_t2 and M_t2 --> Y_t2.

As in any other case, a lagged model requires that you incorporate control variables that may affect X_t1 and M_t2 (and M-->Y vice versa). The synchronous effects model in my point of view is a bit stronger, as X_t1 represents an instrument for X_t2 and, thus, confounding would be reflected in the error covariance (of X_t2 and M_t2, as well as M_t2 and Y_t2).

b) with regard to the moderation part, you can incorporate a latent product term as we illustrated here

Steinmetz, H., Davidov, E., & Schmidt, P. (2011). Three approaches to estimate latent interaction effects: Intention and perceived behavioral control in the theory of planned behavior. Methodological Innovations Online, 6(1), 95-110.

This paper is an application of moderation in a 2-wave panel (Note that the authors applied a more complicated approach to create a product term).

Dormann, C., & Zapf, D. (1999). Social support, social stressors at work, and depressive symptoms: Testing for main and moderating effects with structural equations in a three-wave longitudinal study. Journal of Applied Psychology, 84(6), 874-884.

All the best,

Holger

@Lindsey I guess, Multilevel (longitudinal) moderated mediation analysis can also be done in SPSS using MLMed macro.

https://www.frontiersin.org/articles/10.3389/fpsyg.2020.00637/full