This study explores the dynamic multiscale interactions between carbon and electricity markets during phases I (2005-2007), II (2008-2012) and ongoing III (2013-2016) by introducing the bivariate empirical mode decomposition (BEMD), linear and nonlinear Granger causality tests. Firstly, BEMD is used to decompose the European carbon and electricity prices into several pairs of simple modes with different timescales. Secondly, causality tests are performed at each pair of modes with a similar timescale. Significant but asymmetric causalities can be found that: the bidirectional linear causality turns to a unidirectional one running from the electricity price to the carbon price from phase I to phase III, whereas the nonlinear interaction enhances from insignificance to significance during the three phases. The causality at short-timescale is consistent with that of the original data, while the causalities at mid- and long-timescales are more diversified. The findings obtained show that the effect of carbon market on the electricity market is weakening, which indicates that it is needed to increase its efficiency by raising the carbon price. Moreover, a dynamic adjusting mechanism of emission allowances will be called for as well as the guidance aimed at different timescales in the future.