We analyze the continuous time zero-sum and cooperative controller-stopper games of Karatzas and Sudderth (Ann Probab 29(3):1111-1127, 2001), Karatzas and Zamfirescu (Ann Probab 36(4):1495-1527, 2008) and Karatzas and Zamfirescu (Appl Math Optim 53(2):163-184, 2006) when the volatility of the state process is controlled as in Bayraktar and Huang (SIAM J Control Optim 51(2):1263-1297, 2013) but additionally when the state process has controlled jumps. We perform this analysis by first resolving the stochastic target problems [of Soner and Touzi (SIAM J Control Optim 41(2):404-424, 2002; J Eur Math Soc 4(3):201-236, 2002)] with a cooperative or a non-cooperative stopper and then embedding the original problem into the latter set-up. Unlike in Bayraktar and Huang (SIAM J Control Optim 51(2):1263-1297, 2013) our analysis relies crucially on the Stochastic Perron method of Bayraktar and Sirbu (SIAM J Control Optim 51(6):4274-4294, 2013) but not the dynamic programming principle, which is difficult to prove directly for games.