Laser- and beam-driven plasma accelerators promise electron beam brightness at the exit of plasma cells suitable for X-ray free-electron lasers. Beam transport from the accelerator to the undulator may include a multi-bend, energy-dispersive switchyard, in which energy collimators can be installed to protect the undulator or to serve multiple photon beamlines. Coherent synchrotron radiation and microbunching instability in the switchyard can seriously degrade the brightness of the accelerated beam, reducing the lasing efficiency. We present a semi-analytical analysis of those collective effects for beam parameters expected at the exit of state-of-the-art plasma accelerators. Prescriptions for the linear optics design used to minimize transverse and longitudinal beam instability are discussed.