If there is a single event type that is experienced at most once, the assumptions are the same. Some assumptions just become more obvious in the time-varying situation.
In both what you call "traditional" Cox models (presumably with covariate values all fixed at
time = 0) and in models with time-varying covariates, the modeling is the same. At each event time, the covariate values of the case with the event are compared against those of all cases at risk at that point in time. That's how the regression coefficients for the covariates are determined in both types of models. The only difference with time-varying covariates is that the values an individual contributes to the analysis at one point in time can differ from those the same individual contributes at a different time, whether at that individual's own event time or when the individual is within the risk set for another individual's event.
Thus all the issues of proportional hazards, linearity of log-hazard in the covariates, etc. are the same in both types of models. There might, however, be some practical problems for diagnostics that depend on predicted survival for individuals with time-varying covariates. For example, martingale residuals (difference between observed number of events and those predicted based on the model and follow-up time, used to evaluate the functional form of a continuous-valued covariate) are calculated for each "observation" of a time-varying covariate, leading to multiple residuals for each individual. Which to use for evaluating a functional form? In that case one might use other approaches like spline fitting to determine the proper functional form of the covariate. See Section 5.6 of Therneau and Grambsch. Also, survival software can object to producing predictions based on a set of time-varying coefficient values.
What's very important with time-varying covariates versus a "traditional" Cox model is whether the covariates (1) have been modeled in a way that correctly captures their contributions to the instantaneous risk of the event and (2) avoid the trap of survivorship bias.
For an example of the first point, if kidney failure is the "event," the instantaneous blood glucose or blood pressure or current smoking behavior probably aren't as useful covariates as are measures integrated over time in some way, either endogenously (e.g., hemoglobin A1C as a measure of longer-term blood glucose) or calculated (e.g., pack-years for cigarette smokers). If you don't choose a useful measure of one of these time-varying covariates, one whose value represents the instantaneous risk of an event, you probably won't find a significant coefficient for it in the model. But that's the same thing you would find with a poorly modeled time-invariant covariate.
Ruling out survivorship bias depends on your knowledge of the subject matter. You don't want to include a covariate that can only take on certain values if an individual has already survived a long time without an event. If you do, it's more like the survival time is predicting the covariate value, rather than the other way around. You certainly don't want to include a covariate value that is only determined after the event occurs.