We address the modeling and optimal beamforming (BF) design for multiple-input multiple-output (MIMO) continuous aperture array (CAPA) systems operating over doubly-dispersive (DD) channels. First, a comprehensive DD continuous MIMO (DDC MIMO) channel model that incorporates CAPAs at both the transmitter (TX) and receiver (RX) is derived, which is used to obtain explicit input-output (I/O) relations for various waveforms well suited to integrated sensing and communications (ISAC) and robust to DD channels, namely orthogonal frequency division multiplexing (OFDM), orthogonal time frequency space (OTFS), and affine frequency division multiplexing (AFDM). Then, functional optimization problems are formulated for the design of TX and RX BF matrices that maximize received power, in which novel low-complexity, closed-form solutions are obtained via the calculus of variations (CoV) method, yielding expressions closely related to the classical matched filter commonly used in conventional MIMO systems. Simulation results confirm that the proposed TX/RX BF designs with CAPAs provide significant performance and computational complexity gains over conventional MIMO systems in DD channels.