The Caribbean basin is a geographical area with a high prevalence of asthma due to mineral dust. As such, it is crucial to analyze the dynamic behavior of particulate pollutants in this region. The aim of this study was to investigate the relationships between particulate matter with aerodynamic diameters less than or equal to 2.5 and 10 μm (PM2.5 and PM10) using Hilbert–Huang transform (HHT)-based approaches, including the time-dependent intrinsic correlation (TDIC) and time-dependent intrinsic cross-correlation (TDICC) frames. The study utilized datasets from Puerto Rico from between 2007 and 2010 to demonstrate the relationships between two primary particulate matter concentration datasets of air pollution across multiple time scales. The method first decomposes both time series using improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) to obtain the periodic scales. The Hilbert spectral analysis identified two dominant peaks at a weekly scale for both PM types. High amplitude contributions were sustained for long and continuous time periods at seasonal to intra-seasonal scales, with similar trends in spectral amplitude observed for both types of PM except for monthly and intra-seasonal scales of six months. The TDIC method was used to analyze the resulting modes with similar periodic scales, revealing the strongest and most stable correlation pattern at quarterly and annual cycles. Subsequently, lagged correlations at each time scale were analyzed using the TDICC method. For high-frequency PM10 intrinsic mode functions (IMFs) less than a seasonal scale, the value of the IMF at a given time scale was found to be dependent on multiple antecedent values of PM2.5. However, from the quarterly scale onward, the correlation pattern of the PM2.5-PM10 relationship was stable, and IMFs of PM10 at these scales could be modeled by the lag 1 IMF of PM2.5. These results demonstrate that PM2.5 and PM10 concentrations are dynamically linked during the passage of African dust storms.