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For this, you may have to transform your separated scripts into bash function to be able to build one overall script.

Another way could be to ensure each script won't output anything to STDOUT, than add a cat at end of each script to be able to chain them:

#!/bin/sh

for ((i=1;1<n;i++));do
   read line
   pRoCeSS the $line
   echo >output_log
 done

cat

Final command could look like:

seq 1 10 | cmd1 | cmd2 | cmd2

For using one stream on different process if on same script, you could do:

(
    for ((i=(RANDOM&7);i--;));do
        read line;
        echo CMD1 $line
      done
    for ((i=RANDOM&7;i--;));do
        read line
        echo CMD2 $line
      done
    while read line ;do
        echo CMD3 $line
      done
)
CMD1 1
CMD1 2
CMD1 3
CMD2 4
CMD2 5
CMD2 6
CMD2 7
CMD2 8
CMD2 9
CMD3 10

You may use tee to duplicate command for processing whole stream by many command:

( ( seq 1 10 | tee /dev/fd/5 | sed s/^/line..\ / >&4 ) 5>&1 | wc -l ) 4>&1 
line.. 1
line.. 2
line.. 3
line.. 4
line.. 5
line.. 6
line.. 7
line.. 8
line.. 9
line.. 10
10

or split line by line, using bash:

while read line ;do
    echo cmd1 $line
    read line && echo cmd2 $line
    read line && echo cmd3 $line
  done < <(seq 1 10)
cmd1 1
cmd2 2
cmd3 3
cmd1 4
cmd2 5
cmd3 6
cmd1 7
cmd2 8
cmd3 9
cmd1 10

You may use tee to duplicate command for processing whole stream by many command:

( ( seq 1 10 | tee /dev/fd/5 | sed s/^/line..\ / >&4 ) 5>&1 | wc -l ) 4>&1 
line.. 1
line.. 2
line.. 3
line.. 4
line.. 5
line.. 6
line.. 7
line.. 8
line.. 9
line.. 10
10

or split line by line, using bash:

while read line ;do
    echo cmd1 $line
    read line && echo cmd2 $line
    read line && echo cmd3 $line
  done < <(seq 1 10)
cmd1 1
cmd2 2
cmd3 3
cmd1 4
cmd2 5
cmd3 6
cmd1 7
cmd2 8
cmd3 9
cmd1 10

Finaly there is a way for running cmd1, cmd2 and cmd3 only once with 1/3 of stream as STDIN:

( ( ( seq 1 10 |
         tee /dev/fd/5 /dev/fd/6 |
           sed -ne '1{:a;p;N;N;N;s/^.*\n//;ta;}' |
           cmd1 >&4
     ) 5>&1 |
       sed -ne '2{:a;p;N;N;N;s/^.*\n//;ta;}' |
       cmd2 >&4
  ) 6>&1 |
    sed -ne '3{:a;p;N;N;N;s/^.*\n//;ta;}' |
    cmd3 >&4
) 4>&1 
command_1: 1
command_1: 4
command_1: 7
command_1: 10
Command-2: 2
Command-2: 5
Command-2: 8
command 3: 3
command 3: 6
command 3: 9

For trying this, you could use:

alias cmd1='sed -e "s/^/command_1: /"' \
    cmd2='sed -e "s/^/Command_2: /"' \
    cmd3='sed -e "s/^/Command_3: /"'