File::Find was cool ten years ago, but I've gotten older and lazier; instead, I use File::Find::Rule most of the time. This does what you want:

#!/usr/bin/perl -l

use strict;
use warnings;
use File::Find::Rule;

my $dir = '/tmp';
my @subdirs = File::Find::Rule->in($dir);

my $rule = File::Find::Rule->new;
$rule->file;
$rule->name('*.txt');

my @files = $rule->in(@subdirs);
foreach my $file(@files) {
    print $file;
}
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