require(data.table)
set.seed(333)
t <- data.table(old=1002:2001, dif=sample(1:10,1000, replace=TRUE))
t$new <- t$old + t$dif; t$foo <- rnorm(1000); t$di
It seems to me that you need help from graph algorithms. If you want to start with 1002, you can try:
require(igraph)
g <- graph_from_edgelist(as.matrix(t[,1:2]))
t[old %in% subcomponent(g,"1002","out")]
# 1: 1002 1007 -0.78895338
# 2: 1007 1015 1.13979100
# 3: 1015 1022 -1.21936662
# 4: 1022 1024 1.20390482
# 5: 1024 1026 0.43885860
# ---
#191: 1981 1988 -0.22054875
#192: 1988 1989 -0.22812175
#193: 1989 1995 -0.04687776
#194: 1995 2000 2.41349730
#195: 2000 2002 -1.23425666
Of course you can do the above for each start you want and limiting the results for the first n rows. For instance, we can lapply over the i$start positions and then rbindlist all the values together, declaring an id column with the i$id values. Something like:
n <- 5
rbindlist(
setNames(lapply(i$start, function(x) t[old %in% subcomponent(g,x,"out")[1:n]]), i$id),
idcol="id")
# id old new foo
# 1: 1 1002 1007 -0.7889534
# 2: 1 1007 1015 1.1397910
# 3: 1 1015 1022 -1.2193666
# 4: 1 1022 1024 1.2039048
# 5: 1 1024 1026 0.4388586
# 6: 2 1744 1750 -0.1368320
# 7: 2 1750 1758 0.3331686
# 8: 2 1758 1763 1.3040357
# 9: 2 1763 1767 -1.1715528
#10: 2 1767 1775 0.2841251
#11: 3 1656 1659 -0.1556208
#12: 3 1659 1663 0.1663042
#13: 3 1663 1669 0.3781835
#14: 3 1669 1670 0.2760948
#15: 3 1670 1675 0.3745026