Context. Fast radio bursts (FRBs) are extremely energetic pulses of millisecond duration and unknown origin. To understand the phenomenon that emits these pulses, targeted and un-targeted searches have been performed for multiwavelength counterparts, including the optical. Aims. The objective of this work is to search for optical transients at the positions of eight well-localized (< 1 '') FRBs after the arrival of the burst on different timescales (typically at one day, several months, and one year after FRB detection). We then compare this with known optical light curves to constrain progenitor models. Methods. We used the Las Cumbres Observatory Global Telescope (LCOGT) network to promptly take images with its network of 23 telescopes working around the world. We used a template subtraction technique to analyze all the images collected at differing epochs. We have divided the difference images into two groups: In one group we use the image of the last epoch as a template, and in the other group we use the image of the first epoch as a template. We then searched for optical transients at the localizations of the FRBs in the template subtracted images. Results. We have found no optical transients and have therefore set limiting magnitudes to the optical counterparts. Typical limits in apparent and absolute magnitudes for our LCOGT data are similar to 22 and -19 mag in the r band, respectively. We have compared our limiting magnitudes with light curves of super-luminous supernovae (SLSNe), Type Ia supernovae (SNe Ia), supernovae associated with gamma-ray bursts (GRB-SNe), a kilonova, and tidal disruption events (TDEs). Conclusions. Assuming that the FRB emission coincides with the time of explosion of these transients, we rule out associations with SLSNe (at the similar to 99.9% confidence level) and the brightest subtypes of SNe Ia, GRB-SNe, and TDEs (at a similar confidence level). However, we cannot exclude scenarios where FRBs are directly associated with the faintest of these subtypes or with kilonovae.