  We introduce a general multisummability theory of formal power series in
Carleman ultraholomorphic classes. The finitely many levels of summation are
determined by pairwise comparable, nonequivalent weight sequences admitting
nonzero proximate orders and whose growth indices are distinct. Thus, we extend
the powerful multisummability theory for finitely many Gevrey levels, developed
by J.-P. Ramis, J. \'Ecalle and W. Balser, among others. We provide both the
analytical and cohomological approaches, and obtain a reconstruction formula
for the multisum of a multisummable series by means of iterated generalized
Laplace-like operators.
