How can I interleave rows from 2 data frames together?

How can I interleave rows from 2 data frames together like a perfect riffle shuffle?

Example data:

df1 <- data.frame(df = 1, id = 1:5, chr = 'puppies')
df2 <- data.frame(df = 2, id = 1:5, chr = 'kitties')

df1:

  df id     chr
1  1  1 puppies
2  1  2 puppies
3  1  3 puppies
4  1  4 puppies
5  1  5 puppies

df2:

  df id     chr
1  2  1 kitties
2  2  2 kitties
3  2  3 kitties
4  2  4 kitties
5  2  5 kitties

Desired result:

      df    id     chr
1      1     1 puppies
2      2     1 kitties
3      1     2 puppies
4      2     2 kitties
5      1     3 puppies
6      2     3 kitties
7      1     4 puppies
8      2     4 kitties
9      1     5 puppies
10     2     5 kitties

Assign row numbers to each data frame independently, then bind the rows and sort/arrange by row number and data frame id. In this example, row numbers are trivial since the ids are sequential and act as row number. But in the general case, row numbers should be used.

Here's an example using dplyr:

df1 %>%
  mutate(row_number = row_number()) %>%
  bind_rows(df2 %>% mutate(row_number = row_number())) %>%
  arrange(row_number, df)

Output:

      df    id     chr row_number
   (dbl) (int)   (chr)      (int)
1      1     1 puppies          1
2      2     1 kitties          1
3      1     2 puppies          2
4      2     2 kitties          2
5      1     3 puppies          3
6      2     3 kitties          3
7      1     4 puppies          4
8      2     4 kitties          4
9      1     5 puppies          5
10     2     5 kitties          5