How to load a model with relationships via repository in Magento 2?

Question

Background story:

I was trying to load a model with model_store relationship using this snippet. I failed to do that with my own model. Model is being loaded without store data. Some debuging later I found out that my table_store isn't even joined to main table in queries.

Same problem with a stock model:

I tried same actions with a stock model - \Magento\Cms\Model\Block. Results are the same. I can load an array of blocks but without any store data whatsoever.

Adding a store filter results with a zend error in sql generated by the ORM:

Exception #0 (Zend_Db_Statement_Exception): SQLSTATE[42S22]: Column not found: 
1054 Unknown column 'store_table.store_id' in 'where clause', 
query was: 
SELECT `main_table`.* FROM `cms_block` AS `main_table` WHERE ((`store_table`.`store_id` IN(0, 1, 2, 3, 4, 5))) ORDER BY main_table.block_id ASC

I used following code in a plugin to perform the tests:

<?php
namespace Vendor\Module\Plugin\Checkout\Model\Checkout;

use \Magento\Framework\Api\SearchCriteriaInterface;
use \Magento\Framework\Api\Search\FilterGroup;
use \Magento\Framework\Api\FilterBuilder;

class LayoutProcessor
{
    /**
     * @var SearchCriteriaInterface
     */
    private $criteria;

    /**
     * @var FilterGroup
     */
    private $filterGroup;

    /**
     * @var FilterBuilder
     */
    private $filterBuilder;

    /**
     * LayoutProcessor constructor for DI.
     * @param SearchCriteriaInterface $criteria
     * @param FilterGroup $filterGroup
     * @param FilterBuilder $filterBuilder
     */
    public function __construct(
        SearchCriteriaInterface $criteria,
        FilterGroup $filterGroup,
        FilterBuilder $filterBuilder,
        \Magento\Cms\Model\BlockRepository $blockRepository
    ){
        $this->searchCriteria = $criteria;
        $this->filterGroup = $filterGroup;
        $this->filterBuilder = $filterBuilder;

        $this->filterGroup->setFilters([
            $this->filterBuilder->setField('store')
                ->setConditionType('in')
                ->setValue([0,1,2,3,4,5])
                ->create()
        ]);
        $this->searchCriteria->setFilterGroups([$this->filterGroup]);
        $blocks = $blockRepository->getList($this->searchCriteria)->getItems();
        $firstBlock = $blocks[0];
    }
}

How can I load models with all data from related tables via repository? Currently I'm able to load them correctly only using collection.