In diffusion-based communication, for molecular systems, the achievable data rate depends on the stochastic nature of diffusion, which exhibits a severe inter-symbol-interference (ISI). Multiple-input multiple-output (MIMO) multiplexing improves the data rate at the expense of an inter-link interference (ILI). This paper investigates training-based channel estimation schemes for diffusive MIMO (D-MIMO) systems and corresponding equalization methods. Maximum likelihood and least-squares estimators of mean channel are derived, and the training sequence is designed to minimize the mean square error (MSE). The numerical validations in terms of MSE are compared with Cramér-Rao bound derived herein. Equalization is based on decision feedback equalizer (DFE) structure as this is effective in mitigating diffusive ISI/ILI. Zero-forcing, minimum MSE, and least-squares criteria have been paired to DFE, and their performances are evaluated in terms of bit error probability. D-MIMO time interleaving is exploited as an additional countermeasure to mitigate the ILI with remarkable performance improvements. The configuration of nano-transceivers is not static but affected by a Brownian motion. A block-type communication is proposed for D-MIMO channel estimation and equalization, and the corresponding time-varying D-MIMO MC system is numerically evaluated.