Molecular communication (MC), which allows nanomachines to communicate with each other by using chemical molecules, is considered to be a promising method for communications in liquid environment. Available works on MC mainly focus on modulation and signal detection schemes for MC systems with fixed nanomachines, i.e., fixed molecular communication (FMC) systems. However, the more complex systems with mobile nanomachines (i.e., mobile molecular communication (MMC) systems) have been largely unexplored. This paper considers a MMC system with a fixed transmitter and a mobile receiver communicating over diffusive-drift channels of a limited boundary. We first propose a new modulation scheme to address the issue of misalignment in the signal detection of MMC systems by adopting three types of molecules in the signal modulation and modulating the transmitted signals into blocks with equal length to avoid the transferring of a signal error in the current block on the signal detection in other blocks. We then propose a new signal detection scheme of the MMC systems by calculating the distance between the transmitter and the receiver based on a distance prediction method and detecting signals at the receiver based on the decided adaptive concentration threshold in each time interval. To verify the efficiency of our proposed scheme, we then conducted extensive simulations by the Monte Carlo simulation, and comparisons are also made among our proposed schemes, a well-known fixed threshold signal detection scheme, the CATD scheme, the PAD scheme, and a low complexity signal detection scheme for MMC systems in terms of the BER (bit error rate). Results show that our proposed schemes can outperform these schemes regarding the BER.