After a lot of searching and even questions in the dancer's chat, I decided to write my own extremely simple solution that can be used with Dancer2 easily.

My first Perl cpan module: DBIx::Schema::Migration.

To use it with Dancer2 update your config.yml with:

migrations:
  directory: db/migrations
[download]

Then create db/migrations directory and put the sql migrations files there. All conventions are described in module documentation.

Create migration script: migration.pl

use Dancer2;
use Dancer2::Plugin::Database;
use DBIx::Schema::Migration;

my $dir = config->{migrations}->{directory};

# Instantinating Migration object.
my $migration = DBIx::Schema::Migration->new( {
    dbh => database,
    dir => $dir,
} );

# CLI logic.
my ($action, $num) = @ARGV;

SWITCH: for ($action) { 
    if (/up/)   { $migration->up($num);   last SWITCH; }
    if (/down/) { $migration->down($num); last SWITCH; }
    if (/init/) { $migration->init();     last SWITCH; }
}
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That's all. You can use it via CLI:

perl migration.pl up
perl migration.pl down
perl migration.pl init
[download]