Use curved lines in bumps chart

Using signal::pchip with a grid of X-values works, at least in your example with numeric axes. A proper geom_ would be nice, but hey...

library(tidyverse)
library(signal)
set.seed(47)

df <- as.data.frame(as.table(replicate(8, sample(4))), responseName = 'rank')
df$Var2 <- as.integer(df$Var2)

head(df)
#>   Var1 Var2 rank
#> 1    A    1    4
#> 2    B    1    2
#> 3    C    1    3
#> 4    D    1    1
#> 5    A    2    3
#> 6    B    2    4

ggplot(df, aes(Var2, rank, color = Var1)) +
  geom_line(data = df %>%
              group_by(Var1) %>%
              do({
                tibble(Var2 = seq(min(.$Var2), max(.$Var2),length.out=100),
                       rank = pchip(.$Var2, .$rank, Var2))
              })) +
  geom_point()

Result:

Building on Henrik's answer, this wraps up pchip (I'm using the one from pracma here but the result is the same) so it can be used alongside existing smooth methods more easily:

ggpchip = function(formula, data, weights) structure(pracma::pchipfun(data$x, data$y), class='ggpchip')
predict.ggpchip = function(object, newdata, se.fit=F, ...) {
  fit = unclass(object)(newdata$x)
  if (se.fit) list(fit=data.frame(fit, lwr=fit, upr=fit), se.fit=fit * 0) else fit
}

Then the actual ggplot call is straightforward:

ggplot(df, aes(Var2, rank, color=Var1)) + geom_smooth(method='ggpchip', se=F) + geom_point()

You can then use pchip to smooth other geoms, eg area plots:

ggplot(df, aes(Var2, rank, fill=Var1)) + stat_smooth(method='ggpchip', geom='area', position='fill')