Ride-sourcing drivers enjoy flexibility in scheduling work hours and choosing platforms, which generates multi-homing behavior on multiple platforms. It is challenging to observe the labor supply of multi-homing ride-sourcing drivers due to data limitations, which motivates our research on modeling drivers' dynamic decisions on labor supply in the competitive ride-sourcing market with multiple platforms. We propose a dynamic discrete choice framework by modeling drivers' high-frequency decision sequences on platform switching as high-order hidden Markov processes and considering time-varying factors. The high-order hidden Markov framework relaxes the first-order Markovian assumption and builds interdependencies of unobserved states across multiple time periods. The estimation of the proposed model takes advantage of city-scale multiple platforms datasets in Hangzhou, China, including more than 16 million records of the order payment information and more than 46 thousand records of active ride-sourcing drivers' information. The case study results indicate that the high-order hidden Markov model (HO-HMM) has superior explanatory power in fitting multiple platforms datasets than the multinomial logit model and the first-order hidden Markov model. HO-HMM performs an advantage in modeling the extended historical dependency of drivers' decisions and individual driver behavior modeling with high interpretability. The results uncover the variations of drivers' attitudes in different hidden (unobservable) states and towards different platforms. In general, ride-sourcing drivers respond actively and positively to income and working time. The findings support the platforms' decision-making on pricing, reward, and personalized management of ride-sourcing drivers, and provide beneficial suggestions for improving income in the competitive ride-sourcing market.