name=stackexchange

#!/bin/bash
source ./variables.txt
echo $name

ssh -i "auth.key" user@host 'bash -s' < test.sh

bash: line 2: ./variables.txt: No such file or directory

ssh -i "auth.key" user@host 'bash -s' < test.sh

bash: line 2: ./variables.txt: No such file or directory

name=remotehost

<test.sh sed '/^[ t]*source variables.txt[ t]*$/ {
  d
  r variables.txt
}' | ssh -i "auth.key" user@host 'bash -s'