Fluctuation-based dispersion entropy (FDispEn) is a new approach to estimate the dynamical variability of the fluctuations of signals. It is based on Shannon entropy and fluctuation-based dispersion patterns. To quantify the physiological dynamics over multiple time scales, multiscale FDispEn (MFDE) is developed in this article. MFDE is robust to the presence of baseline wanders, or trends, in the data. We evaluate MFDE, compared with popular multiscale sample entropy (MSE), and the recently introduced multiscale dispersion entropy (MDE), on selected synthetic data and five neurological diseases' datasets: 1) focal and non-focal electroencephalograms (EEGs); 2) walking stride interval signals for young, elderly, and Parkinson's subjects; 3) stride interval fluctuations for Huntington's disease and amyotrophic lateral sclerosis; 4) EEGs for controls and Alzheimer's disease patients; and 5) eye movement data for Parkinson's disease and ataxia. MFDE dealt with the problem of undefined MSE values and, compared with MDE, led to more stable entropy values over the scale factors for pink noise. Overall, MFDE was the fastest and most consistent method for the discrimination of different states of neurological data, especially where the mean value of a time series considerably changes along the signal (e.g., eye movement data). This study shows that MFDE is a relevant new metric to gain further insights into the dynamics of neurological diseases recordings.