  In this paper, we prove that the ratio of the modulus of the iterates of two
points in an escaping Fatou component may be bounded even if the orbit of the
component contains an infinite modulus annulus sequence and this case cannot
happen when the maximal modulus of the meromorphic function is large enough.
Therefore, we extend the related results for entire functions to ones for
meromorphic functions with infinitely many poles. And we investigate the fast
escaping Fatou components of meromorphic functions defined in [44] in terms of
the Nevanlinna characteristic instead of the maximal modulus in [12] and show
that the multiply-connected wandering domain is a part of the fast escaping set
under a growth condition of the maximal modulus. Finally we give examples of
wandering domains escaping at arbitrary fast rate and slow rate.
