Extragalactic transients, such as gamma-ray bursts (GRBs), represent the most energetic phenomena in the Universe. They are characterized by their intense radiation within short timescales. In recent years, significant advances have been made in understanding the emission mechanism of GRBs, with the detection of very-high-energy (VHE; >100 GeV) gamma rays. These observations have improved our understanding of VHE gamma-ray emissions in such extreme environments. So far, five GRBs have been detected in VHE gamma rays, intensifying the search for more such detections. The observation of extragalactic sources in VHE gamma rays is challenging due to extragalactic background light (EBL) absorption, which results in softening the source spectrum to energies below a few TeV.

To address these challenges, the ALTO/CoMET R&D project is proposed. This project aims to design a wide field-of-view observatory to observe soft-spectrum sources continuously in VHE gamma rays. To achieve this goal, a dedicated Monte Carlo simulation study is conducted to optimize the configuration of the proposed detectors. An important part of the simulation study is the development of an analysis method called SEMLA (Signal Extraction using Machine learning for ALTO/CoMET), which enables the proposed array to detect gamma rays at energies down to a few hundred GeV. The results from the simulated data analysis demonstrate that ALTO/CoMET is capable of detecting extragalactic transients like GRBs. 

This thesis presents the analysis methods developed, and the results obtained from the ALTO/CoMET R&D project in detecting extragalactic transients, with a focus on GRBs. The lessons learned from these ALTO/CoMET simulation studies and prototype activities are useful for other future extensive air shower detectors, concerning the observation of soft-spectrum VHE gamma-ray sources.

Additionally, the search for GRB emission is explored using the H.E.S.S. array, analysing follow-up observations of GRBs recorded from its construction in 2004 until 2019 (excluding the two significantly detected GRBs by H.E.S.S.).