You should be able to do this with python scripting.
You should be able to have python create a new file geodatabase.
Then use ListFeatureClasses from your first file geodatabase to copy this data into your new file geodatabase.
Then use ListListFeatureClasses from your second file geodatabase to append this data into your new file geodatabase onto the feature classes that were created when the data from the first file geodatabase was copied into the new file geodatabase.
You might consider posting this thread in the python subforum.
Kind of, you can adapt it to your particular case.
import arcpy
arcpy.env.overwriteOutput = True
try:
ws = arcpy.env.workspace = "your workspace"
a = arcpy.ListWorkspaces("", "Access")
print a
for x in a:
b = arcpy.env.workspace = x
c = arcpy.ListTables()
for y in c:
out = "destination folder"
arcpy.TableToDBASE_conversion(y, out)
except:
print "Se ha producido un error"
print(arcpy.GetMessages(2))
You can also do this kind of thing with Model Builder. In fact it's awfully handy for simple but cumbersome once-off tasks like this. Advanced iteration in Model Builder can get complicated and takes some skill, but you don't have to write code.