We present a detailed study of the carrier-envelope offset (CEO) frequency dynamics of SESAM modelocked thin disk lasers (TDLs) pumped by kW-class highly transverse multimode pump diodes with a typical M(2) value of 200-300, and give guidelines for future frequency stabilization of multi-100-W oscillators. We demonstrate CEO frequency detection with > 30 dB signal-to-noise ratio with a resolution bandwidth of 100 kHz from a SESAM modelocked Yb:YAG TDL delivering 140 W average output power with 748-fs pulses at 7-MHz pulse repetition rate. We compare with a low-power CEO frequency stabilized Yb:CALGO TDL delivering 2.1 W with 77-fs pulses at 65 MHz. For both lasers, we perform a complete noise characterization, measure the relevant transfer functions (TFs) and compare them to theoretical models. The measured TFs are used to determine the propagation of the pump noise step-by-step through the system components. From the noise propagation analysis, we identify the relative intensity noise (RIN) of the pump diode as the main contribution to the CEO frequency noise. The resulting noise levels are not excessive and do not prevent CEO frequency stabilization. More importantly, the laser cavity dynamics are shown to play an essential role in the CEO frequency dynamics. The cavity TFs of the two lasers are very different which explains why at this point a tight CEO frequency lock can be obtained with the Yb:CALGO TDL but not with the Yb:YAG TDL. For CEO stabilization laser cavities should exhibit high damping of the relaxation oscillations by nonlinear intra-cavity elements, for example by operating a SESAM in the roll-over regime. Therefore the optimum SESAM operation point is a trade-off between enough damping and avoiding multiple pulsing instabilities. Additional cavity components could be considered for supplementary damping independent of the SESAM operation point.