One way would be to find the name of the column in the dataframe and split based on that. We can use group_split to split data into dataframes, we use map_at to remove the 1st row from each list since it is the column name and use type.convert to convert columns into it's respective classes. 
library(dplyr)
library(purrr)
temp <- df %>%
          group_split(id = cumsum(A == names(.)[1]) + 1) %>%
          map_at(-1, tail, -1) %>%
          map(type.convert)
temp
#[[1]]
# A tibble: 3 x 5
#     A     B     C     D     id
#   <dbl> <dbl> <dbl> <dbl> <int>
#1 0.668 0.411 0.553 0.477     1
#2 0.794 0.821 0.530 0.732     1
#3 0.108 0.647 0.789 0.693     1
#[[2]]
# A tibble: 6 x 5
#    A     B     C     D    id
#  <dbl> <dbl> <dbl> <dbl> <int>
#1 0.724 0.783 0.023 0.478     2
#2 0.861 0.099 0.407 0.332     2
#3 0.438 0.316 0.913 0.651     2
#4 0.245 0.519 0.294 0.258     2
#5 0.07  0.662 0.459 0.479     2
#6 0.766 0.839 0.892 0.961     2
#[[3]]
# A tibble: 4 x 5
#      A     B     C     D    id
#  <dbl> <dbl> <dbl> <dbl> <int>
#1 0.084 0.347 0.864 0.435     3
#2 0.875 0.334 0.39  0.713     3
#3 0.339 0.476 0.777 0.4       3
#4 0.084 0.347 0.864 0.435     3
Using the same logic in base R, we can do 
df$id <- cumsum(df$A == names(df)[1]) + 1
temp <- split(df, df$id)
temp[-1] <- lapply(temp[-1], tail, -1)
temp <- lapply(temp, type.convert)
If you need them as separate dataframes do,
names(temp) <- paste0("df", seq_along(temp))
list2env(temp, .GlobalEnv)
data
df <- structure(list(A = structure(c(7L, 10L, 3L, 13L, 8L, 11L, 6L, 
4L, 1L, 9L, 13L, 2L, 12L, 5L, 2L), .Label = c("0.070", "0.084", 
"0.108", "0.245", "0.339", "0.438", "0.668", "0.724", "0.766", 
"0.794", "0.861", "0.875", "A"), class = "factor"), B = structure(c(5L, 
11L, 8L, 13L, 10L, 1L, 2L, 7L, 9L, 12L, 13L, 4L, 3L, 6L, 4L), .Label = c("0.099", 
"0.316", "0.334", "0.347", "0.411", "0.476", "0.519", "0.647", 
"0.662", "0.783", "0.821", "0.839", "B"), class = "factor"), 
 C = structure(c(7L, 6L, 9L, 13L, 1L, 4L, 12L, 2L, 5L, 11L, 
13L, 10L, 3L, 8L, 10L), .Label = c("0.023", "0.294", "0.390", 
"0.407", "0.459", "0.530", "0.553", "0.777", "0.789", "0.864", 
"0.892", "0.913", "C"), class = "factor"), D = structure(c(5L, 
11L, 9L, 13L, 6L, 2L, 8L, 1L, 7L, 12L, 13L, 4L, 10L, 3L, 
4L), .Label = c("0.258", "0.332", "0.400", "0.435", "0.477", 
"0.478", "0.479", "0.651", "0.693", "0.713", "0.732", "0.961", 
"D"), class = "factor")), class = "data.frame", row.names = c(NA, -15L))