This paper addresses an adaptive blind multiuser detection strategy for a multi-rate direct sequence CDMA downlink channel. Specifically, two chip-level equalisation schemes, using either variable spreading length (VSL) or multi-code (MCD) multi-rate access modes, are proposed and tested. Both equalisers can be updated by minimising a hybrid CM/MSE cost function based on the constant modulus (CM) criterion for active users and a mean square error (MSE) criterion for inactive users in a partially loaded system. The BER performance abd the convergence of the proposed algorithms are analysed and compared through various simulations in both partially and fully loaded systems. Essentially, our results show that the VSL algorithm exhibits faster convergence than the MCD scheme while similar BER performance over different multi-path channels is achieved.