We report on a search for persistent radio emission from the one-off Fast Radio Burst (FRB) 20190714A, as well as from two repeating FRBs, 20190711A and 20171019A, using the MeerKAT radio telescope. For FRB 20171019A we also conducted simultaneous observations with the High Energy Stereoscopic System (H.E.S.S.) in very high energy gamma rays and searched for signals in the ultraviolet, optical, and X-ray bands. For this FRB, we obtain a UV flux upper limit of 1.39x10^-16 erg/cm^-2/s/Amstrong, X-ray limit of ~ 6.6x10^-14 erg/cm^-2/s and a limit on the very-high-energy gamma-ray flux (Phi) (E > 120 GeV) < 1.7 x 10^-12 erg/cm^-2/s. We obtain a radio upper limit of ~15 microJy/beam for persistent emission at the locations of both FRBs 20190711A and 20171019A, but detect diffuse radio emission with a peak brightness of ~53 microJy/beam associated with FRB 20190714A at z = 0.2365. This represents the first detection of the radio continuum emission potentially associated with the host (galaxy) of FRB 20190714A, and is only the third known FRB to have such an association. Given the possible association of a faint persistent source, FRB 20190714A may potentially be a repeating FRB whose age lies between that of FRB 20121102A and FRB 20180916A. A parallel search for repeat bursts from these FRBs revealed no new detections down to a fluence of 0.08 Jy ms for a 1 ms duration burst.