For example, the input here has two columns (and zero rows), but the output has only one column:

dat <- data.frame(x = numeric(0), g = character(0))
dat %>% group_by(g) %>% do(identity(.)) %>% str()
# Classes ‘grouped_df’, ‘tbl_df’, ‘tbl’ and 'data.frame':  0 obs. of  1 variable:
#  $ g: Factor w/ 0 levels: 
#  - attr(*, "vars")=List of 1
#   ..$ : symbol g
#  - attr(*, "drop")= logi TRUE
#  - attr(*, "indices")= list()
#  - attr(*, "group_sizes")= int 
#  - attr(*, "biggest_group_size")= int 0
#  - attr(*, "labels")='data.frame':    0 obs. of  1 variable:
#   ..$ g: Factor w/ 0 levels: 
#   ..- attr(*, "vars")=List of 1
#   .. ..$ : symbol g

In my particular case, I would like it to simply return the input unchanged, but I'm not sure what the correct output should be, in general.

Since the function isn't actually called on zero-row data, there's no way of knowing what columns it would return. Maybe it should be called once with the zero-row data frame? In this case, that would be equivalent to doing dat %>% group_by(g) %>% identity().