Multinomial regression for a multi-state survival problem has the same limitation that binomial regression has for single-event survival models: it doesn't nicely handle censored survival times.

Strict competing risks models assume that all non-initial states are absorbing states. If you are modeling multiple disease stages or back-and-forth transitions among states, that won't work.

That leaves multi-state models, which can take several forms. The R "Multi-state models and competing risks" vignette is a good introduction. For panel data like you seem to have (same individuals followed over a set of discrete time periods), Section 6 of the vignette recommends the msm package.

A multinomial regression at a single evaluation time for a multi-state survival problem has the same limitation that a single binomial regression has for single-event survival models: it doesn't nicely handle censored survival times prior to the single evaluation time. Treating "censored" (loss to follow up) as a separate state poses an interesting problem: if censoring can be predicted by the same covariates that are associated with outcome, then censoring is presumably informative and the standard assumption of non-informative censoring in survival analysis doesn't hold. You would have to do more extensive modeling.

I don't see that separate single multinomial regressions based on each of the non-absorbing states, as suggested in the question, would be appropriate.

You can, however, analyze discrete-time survival data as a series of binomial, ordinal, or multinomial regressions at each of the observation times. The data need to be formatted properly, so that only those who are still at risk are included in the regression at an observation time. Censoring is then handled by the omission of individuals without an event time from regressions at times after the censoring time. Tutz and Schmid describe the rationale. The discSurv package in R provides tools for formatting data in a way that allows application of standard regression methods to discrete-time data.

Strict competing-risks models assume that all non-initial states are absorbing states. If you are modeling multiple disease stages or back-and-forth transitions among states, a competing-risks model won't work. Tutz and Schmid only seem to describe multinomial models for competing risks scenarios; I don't think that they cover back-and-forth transitions between states at all.

That leaves multi-state models, which can take several forms.* You might treat such a model as a series of multinomial models in time, as in the example you cite in a comment, but if back-and-forth transitions are allowed between states (unlike the unidirectional transitions in that example) that could become messy. You might as well use Markov-type transition models that are designed for such data.

The R "Multi-state models and competing risks" vignette is a good introduction. For panel data like you seem to have (same individuals followed over a set of discrete time periods), Section 6 of the vignette recommends the msm package. I don't regularly deal with discrete-time survival data and I don't have experience with that package, however, so I can't provide an informed opinion about its suitability.

*Even a simple standard survival model can be considered "multi-state" if you allow for two states (initial state and absorbing event state) to be called "multi." To that extent, some of this discussion is terminological rather than substantive.