  Open clusters are known as excellent tools for various topics in Galactic
research. For example, they allow accurately tracing the chemical structure of
the Galactic disc. However, the metallicity is known only for a rather low
percentage of the open cluster population, and these values are based on a
variety of methods and data. Therefore, a large and homogeneous sample is
highly desirable. In the third part of our series we compile a large sample of
homogenised open cluster metallicities using a wide variety of different
sources. These data and a sample of Cepheids are used to investigate the radial
metallicity gradient, age effects, and to test current models. We used
photometric and spectroscopic data to derive cluster metallicities. The
different sources were checked and tested for possible offsets and
correlations. In total, metallicities for 172 open cluster were derived. We
used the spectroscopic data of 100 objects for a study of the radial
metallicity distribution and the age-metallicity relation. We found a possible
increase of metallicity with age, which, if confirmed, would provide
observational evidence for radial migration. Although a statistical
significance is given, more studies are certainly needed to exclude selection
effects, for example. The comparison of open clusters and Cepheids with recent
Galactic models agrees well in general. However, the models do not reproduce
the flat gradient of the open clusters in the outer disc. Thus, the effect of
radial migration is either underestimated in the models, or an additional
mechanism is at work. [abridged]
