Assessing mediation with lagged predictors

Hello dear forum members!

I am seeking your comments/advises on the following question related to assessing mediation effects. Following the general guidelines by Baron and Kenny (1986):

1. y(it) = a + x(it) + u --> significant impact
2. m(it) = a + x(it) + u (where m -- mediator) --> significant impact
3. y(it) = a + x(it) + m(it) + u --> significant impact of m and greatly reduced impact of x (suggesting partial mediation)

Note, however, that my data is panel and theory suggests that x(it-1) is also a predictor of y. So, I add it into the model and assess again:

1. y(it) = a + x(it) + x(it-1) + u --> significant impact of both regressors
2. m(it) = a + x(it) + x(it-1) + u --> Question: Do I have to include both x's simultaneously OR separately? In the former case, only x(it) is significant. In the latter case, each has a significant impact by itself. My gut feeling is that both have to be tested simultaneously (i.e., former case).

3. y(it) = a + x(it) + x(it-1)+ m(it) + u --> impact of x(it) becomes insignificant -- suggesting full mediation; impact of x(it-1) is still significant -- suggesting no mediation.

I further use the Sobel test to test the significance of the suggested mediation effects.

Please suggest if my logic here is plausible or am I missing something? Your comments are greatly appreciated.
Here is a fantastic paper on the topic: Selig, J. P., & Preacher, K. J. (2009). Mediation models for longitudinal data in developmental research. Research in Human Development, 6(2-3), 144-164.

Have a good week end, y'all :)