Multilevel models usually assume that the effect at level 1 and the effect at level 2 of a level-1 variable are equal.  If you want to allow for different level 1 and level 2 (and, potentially, level 3) effects then you can include the group mean of the level-1 variable as you would any other higher level variable.  I don't use HLM but I assume this is relatively doable.

See Snijders and Bosker 2012 p56-60, and also this working paper:
http://polmeth.wustl.edu/mediaDetail.php?docId=1324

Hope that helps!

Thank you for your quick response. The references you suggest are very useful for accounting for the between effect of level-1 time-varying predictor/covariate by adding the higher-level mean into the model of the higher level.

My question is more about how to deal with the effect of level-2 or higher-level time-varying variables.  Usually, the level-2 or higher-level variable is time-invariant (e.g. students within schools, the number of enrollment in school is regarded as constant over time, and it is easy to incorporate its effect in the level-2 model). However, those variables could vary over time. For example, given a hierarchical structure like households within neighborhoods, the characteristics of neighborhoods are dynamic and can change over time. It seems not appropriate to add those variables in the level-2 or 3 (neighborhood level); but if we add them to the level-1 (repeated measures for households), we also need to include dummy variables for the neighborhood entity.

In my mind, the combined model would be formulated as follows.
Ytij = a0 + a1*X'tij + a2*Z'tj + Utj + Etij

t  (occasions) - represent the level-1; i (individuals) for level-2, j (neighborhood) for level-3;
X'  is level-2 time-varying predictor,
Z' is level-3 time-varying predictor;

Is there any multilevel model specification that can incorporate the trajectory of individuals as well as the trajectory of place-based groups such as neighborhoods or census tracts? Could you provide a reference explaining or examplifying such a model?

Assuming that your time points are in the level-one portion of the model you can include the time-varying covariates there regardless of their association with a higher level characteristic.

Yes, my time points are in the level-one portion of the model. But both of lower and higher level characteristics change over time. I want to add the time-varying covariates that represent higher level characteristic into the multilevel model. Can I achieve it? If yes, how?

If I understand correctly, I think what you need is a four-level model, where occasions (t) are cross-classified in individuals (i) and in area-waves (w), and area-waves are in turn nested in areas (j). Individuals may also be nested in areas, if they never move from one area to another.

Per Andy's suggestion, you could mean-center both the individual- and area-level variables that are time-varying, yielding mean components indexed i and j, and mean-centered (or de-meaned) components t and w.

I'm not aware of any applications that have used such an approach, though there may be something out there. I know that R's lme4 and MCMCglmm packages, and MLwiN can handle four-level models of this kind -- whether other packages can I don't know. However, no matter the software, this is getting to be quite a complicated model to interpret, and will make heavy demands on the data.

Aside from the papers Andy suggested, you can also refer http://seis.bris.ac.uk/~ggmhf/MHF.MLM-longit.2013.pdf.

This addresses the analysis of repeated cross-sectional survey data, where repeated surveys are made of the same areas (e.g., countries) over time, but the individuals change across waves. That's a three-level situation. In your case, as mentioned above, you have an additional level, because individual people are observed multiple times.