The thing that motivates 'a company' (meaning, as a reminder, the Board of Directors voted in by the majority of voting shareholders) to pay dividends to non-cumulative preferred shareholders (which are often non-voting), is that preferred shares get preference in the order of dividends being paid out.
If I own 100% of the voting common shares of my private company, and you own non-voting non-cumulative preferred shares in that same company, I can't pay myself dividends until you get yours. Your preferred shares may have a clause stating what your owed annual amount is, even if it is not cumulative year over year. So if you own 100,000 shares with a stated dividend of $1, I can't pay myself a penny of dividends as a common shareholder, until in that same year I pay you your owed $100,000.
There are also typically jurisdictional laws that protect minority shareholders from mismanagement by majority shareholders, such that if you tried to do something like liquidate the company without paying me my owed dividends, I might be able to sue for payment. In reality however, non-cumulative preferred shares do have a limited value because if the example company above earns $1M a year, and you expect to effectively get 10% of its annual earnings through your dividends, I could simply not pay dividends for 5 years, then pay you $100k and pay myself the remaining $4.9M.