The relativistic klystron amplifier (RKA) is one of the most efficient sources to amplify a high-power microwave signal due to its intrinsic merit of high-power conversion efficiency, high gain and stable operating frequency. However, the transverse dimensions of the RKA dramatically decrease when the operating frequency increases to X band, and the power capacity of the RKA is limited by the transverse dimensions. An X-band multiple-beam relativistic klystron amplifier is proposed to overcome the radiation power limitation. Each electron beam propagates in separate drift tubes and shares the same coaxial interaction cavities in the multiple-beam relativistic klystron amplifier, and the transverse dimensions of the multiple-beam relativistic klystron amplifier are free from the operating frequency restriction and a microwave power of over 1 GW is generated in the experiment. For a high-power electron device, the transmission of electron beam is critical, and the power conversion efficiency of the device is affected. In this paper, we conduct an investigation into the transmission process of the intense relativistic multiple electron beams, and the number of the multiple electron beams is set to be 16. It is found that when the multiple electron beam is transmitted in the device, the electron beam rotates around the center of the whole device, causing the electron beam to deviate from the drift tube channel. At the same time, each electron beam rotates around itself, and the cross section of the electron beam is deformed and expanded. In the improper design of electron beam and drift tube parameters, two kinds of rotating motions cause beam to lose. A multiple-electron-beam diode structure is optimized by the particle-in-cell simulation to reduce beam loss, with the effects of the related factors taken into account. Each pole of the cathodes is made up of graphite and stainless steel. The cathode head is made up of graphite, for the graphite has a lower emission threshold. The cathode base and cathode pole are made up of stainless steel, for the stainless steel has a higher emission threshold. Also the shape and structure of cathode pole, cathode head and anode are optimized to reduce the electric field intensity on the cathode pole and enhance the electric field intensity on the end face of cathode head. At the same time, the electric field distribution of the cathode head is uniform to improve the electron beam emission uniformity. The simulation results demonstrate that the transmission efficiency of multiple electron beams can reach 99%. In the experiment, the transmission efficiency of multiple electron beams is 92% with a beam voltage and beam current of 801 kV and 9.3 kA, respectively.