Multinomial logit / time-series fixed effects / multivariate regression: Which one?

As part of a larger study, we have collected a wealth of data on the interactions customers engage in when buying and using a service.

Particularly, we have distinguished this process into 8 phases that tend to - but are not necessarily - sequential. Also, not all customers experience all 8 phases, leading to only 1/3 of the participants (total of 3,015) to answer all questions.

For each of these phases we have asked participants for their main interaction partner („channel“, 12 different options) and means of communication used (how the interaction happened, e.g., via telephone or online, in total 9 different options). For some combinations between the two, e.g., website (=channel) and PC/laptop (=means) high correlations exist, so that these two variables cannot be assumed to be independent.

Based on a large qualitative study we ran on this exact topic as well as the results of descriptive statistics, we have a number of strong hypotheses on the factors that influence customers‘ interaction choices (channels and means), such as what interactions customers picked in the previous phase, how they liked previous interactions, the extent of their product/service knowledge etc.

Now, I am unsure about the correct statistical test to use. Any recommendations would be much appreciated!