@Emmanuel.W Oh I should have read you very first paragraph As a preamble, I would like to make it clear that I am talking about model with time-dependent covariates and not with time-dependent coefficients. I guess this is harder for the semiparametric cox model since you only have a partial likelihood.
If your task is prediction then you can use cross validation or similar procedures to see if you get an improvement in e.g. outsample concordance. Further, your are not adding more coefficients so you can do it in-sample without adjustments as far as I see.
You may find this post relevant. The models with and without time-varying covariates are not nested so I guess it boils downs to model selection with two equally complicated models.
If you are using a Cox model in R then you can use the
survival::cox.zph which test a correlation between the a transformation and the scaled Schoenfeld residuals. You can see the details in the article on the help page
?survival::cox.zph. You have to be aware that you are testing a correlation coefficient so non-monotonic may not be captured by the test (if I remember correctly).
There are other options. See e.g. the test in:
Therneau, T. M., Grambsch, P. M., & Fleming, T. R. (1990). Martingale-based residuals for survival models. Biometrika, 77(1), 147-160.
or the 19.5.2 Proportional Hazards Assumption in:
Harrell Jr, F. E. (2015). Regression modeling strategies: with applications to linear models, logistic and ordinal regression, and survival analysis. Springer.