Summary: Exposures to environmental chemicals during gestation can alter health status later in life. Most studies of maternal exposure to chemicals during pregnancy have focused on a single chemical exposure observed at high temporal resolution. Recent research has turned to focus on exposure to mixtures of multiple chemicals, generally observed at a single time point. We consider statistical methods for analyzing data on chemical mixtures that are observed at a high temporal resolution. As motivation, we analyze the association between exposure to four ambient air pollutants observed weekly throughout gestation and birth weight in a Boston-area prospective birth cohort. To explore patterns in the data, we first apply methods for analyzing data on: (1) a single chemical observed at high temporal resolution, and (2) a mixture measured at a single point in time. We highlight the shortcomings of these approaches for temporally-resolved data on exposure to chemical mixtures. Second, we propose a novel method, a Bayesian kernel machine regression distributed lag model (BKMR-DLM) that simultaneously accounts for nonlinear associations and interactions among time-varying measures of exposure to mixtures. BKMR-DLM uses a functional weight for each exposure that parameterizes the window of susceptibility corresponding to that exposure within a kernel machine framework that captures nonlinear and interaction effects of the multivariate exposure on the outcome. In a simulation study we show that the proposed method can better estimate the exposure-response function and, in high signal settings, can identify critical windows in time during which exposure has an increased association with the outcome. Applying the proposed method to the Boston birth cohort data, we find evidence of a negative association between organic carbon and birth weight and that nitrate modifies the organic carbon, elemental carbon, and sulfate exposure-response functions.