How to create a new column by appending values to a list from other columns in R

I would like to make a new column by appending to a list conditional on the values of other columns. If possible, I would like to do so in dplyr. Sample input and desired output is below.

Suppose a dataframe newdata:

col1 col2 col3 col4
dog  cat  NA   NA
NA   cat  foo  bar
dog  NA   NA   NA
NA   cat  NA   NA

Here is my desired output, with the new column newCol:

col1 col2 col3 col4 newCol
dog  cat  NA   NA   (dog, cat)
NA   cat  foo  bar  (cat, foo, bar)
dog  NA   NA   NA   (dog)
NA   cat  NA   bar  (cat, bar)

I have tried using ifelse within mutate and case_when within mutate, but both will not allow concatenation to a list. Here is my (unsuccessful) attempt with case_when:

newdata = newdata %>% mutate( 
    newCol = case_when(
        col1 == "dog" ~ c("dog"),
        col2 == "cat" ~ c(newCol, "cat"),
        col3 == "foo" ~ c(newCol, "foo"),
        col4 == "bar" ~ c(newcol, "dog")
        )
    )

I tried a similar approach with an ifelse statement for each column but also could not append to the list. Any help would be much appreciated.