data.table::foverlaps + foreach + iterators

library(foreach)
library(iterators)
library(data.table)
library(pipeR)

set.seed(10)
k <- 1
outList <- foreach(v = iter(matrix(sample(3:29, 6000, TRUE), 1000), by = "row")) %:% when(k <= 4) %do%
  {
    if (all(diff(sort(v)) > 2)) {
      k <- k + 1
      return(data.table(studentID = k, matrix(c(1, sort(v), 31), 4, 2, TRUE, list(NULL, c("Start", "End")))))
    } else return(NULL)
  }
outDT <- rbindlist(outList) %>>% `[`(j = `:=`(studentID = match(studentID, sort(unique(studentID))),
                                              avgScore = abs(rnorm(nrow(.)))))
#     studentID Start End  avgScore
#  1:         1     1   3 0.4605151
#  2:         1     6  10 0.2350253
#  3:         1    19  22 0.6432573
#  4:         1    25  31 0.9131981
#  5:         2     1   4 0.9882860
#  6:         2     7  11 0.1127413
#  7:         2    16  20 1.4900499
#  8:         2    26  31 0.4432356
#  9:         3     1   5 1.3623441
# 10:         3    10  14 1.0452357
# 11:         3    21  25 0.2339315
# 12:         3    28  31 2.5524180
# 13:         4     1   4 1.7687187
# 14:         4     7  10 0.6595706
# 15:         4    19  23 0.3707332
# 16:         4    26  31 0.5928033

iter <- isplit(outDT, outDT$studentID)
resDT <- copy(iter$nextElem()$value) %>>% `[`(j = `:=`(studentID = NULL))
setkey(resDT, Start, End)
while (TRUE) {
  v <- tryCatch(iter$nextElem(), error = function(e) e)
  if (any(class(v) == "error"))
    break
  resDT <- foverlaps(v$value, resDT, type = "any", nomatch = 0) %>>%
    `[`(j = `:=`(Start = pmax(Start, i.Start), End = pmin(End, i.End))) %>>%
    `[`(j = .(Start, End))
  setkey(resDT, Start, End)
}
#    Start End
# 1:     1   3
# 2:    10  10
# 3:    28  31

finalResDT <- foreach(it = isplit(outDT, outDT$studentID), .final = rbindlist) %do%
  {
    foverlaps(it$value, resDT, type = "any", nomatch = 0) %>>%
      `[`(j = avgScore := (i.End - End + 1) / (Start - i.Start + 1) * avgScore) %>>%
      `[`(j = .(Start, End, studentID, avgScore))
  } %>>% dcast(Start + End ~ studentID, val.var = "avgScore") %>>%
  setnames(as.character(1:(ncol(.)-2)), paste0("studentID-", 1:(ncol(.)-2)))
#    Start End studentID-1 studentID-2 studentID-3 studentID-4
# 1:     1   3  0.46051506  1.97657201    4.087032   3.5374375
# 2:    10  10  0.04700506  0.05637067    5.226179   0.1648927
# 3:    28  31  0.22829953  0.14774520    2.552418   0.1976011