Both internalization (sometimes with degradation) and changes in gene expression can occur; the circumstances leading to the down regulation determine which (or both). It isn't necessary for receptors to be bound to their ligand to be internalized, though, and it isn't the internalization of receptors that causes changes in gene expression (I suppose it is possible that happens someplace, but not that I am familiar with). The processes involve activation of second messenger systems and kinases that either phosphorylate/dephosphorylate the receptors to mark them for internalization and/or degradation, or activate/deactivate transcription factors.
Here are two reviews with more information: one is newer but focused on a single class of receptors and fairly in-depth; the other is a bit dated but more broad, but brief:
Williams, J. T., Ingram, S. L., Henderson, G., Chavkin, C., von Zastrow, M., Schulz, S., ... & Christie, M. J. (2013). Regulation of µ-opioid receptors: Desensitization, phosphorylation, internalization, and tolerance. Pharmacological reviews, 65(1), 223-254.
Tsao, P., & von Zastrow, M. (2000). Downregulation of G protein-coupled receptors. Current opinion in neurobiology, 10(3), 365-369.
You can also read about long term depression, which is different from homeostatic responses to ligand concentrations, but it does seem to serve a homeostatic function to keep overall synaptic strengths from growing indefinitely, and the molecular pathways are very similar (and it is a very well-studied phenomenon so you will find a lot of accessible information).