  We study the spaces of Besov and Triebel-Lizorkin type with variable
smoothness and integrability as introduced recently by Almeida & H\"ast\"o and
Diening, H\"ast\"o & Roudenko. Both scales cover many classical spaces with
fixed exponents as well as function spaces of variable smoothness and function
spaces of variable integrability. These spaces have been introduced by Fourier
analytical tools, as the decomposition of unity. Surprisingly, our main result
states that these spaces also allow a characterization in the time-domain with
the help of classical ball means of differences.
  To that end, we first prove a local means characterization for them with the
help of the so-called Peetre maximal functions. Our results do also hold for
2-microlocal function spaces with variable integrability which are a slight
generalization of generalized smoothness spaces and spaces of variable
smoothness.
