What are the best ways to manage part time remote staff on equity?

You could:
1. make a connection between per project and equity allocation - so each time they complete a project, they get a small % of equity. Research has shown that people prefer small payouts than one big payout at the end (ppl can't postpone their desires...).
2. Keep them involved in the decision loop and share with them small successes. Make them feel part of the company/business.
3. Give out additional 'bonuses' for projects done well or for milestones achieved by the company.
4. If you have the capital/money to support it, then make it clear that they can exchange their equity for money - if a company is willing to buy back it's equity, this shows that the equity has value - which will make them want to keep it. The only challenge with this last option is that if you do buy back shares, it creates a tax event for both sides which isn't always advised.
There are a few additional options.
I've successfully helped over 300 entrepreneurs and would be happy to help you. After scheduling a call, be sure to please send me some background information so that I can prepare before - thus giving you maximum value for your money during the call itself.

When you say 'on equity' do you mean you have an equity pool you're using as motivation? And you're experiencing less motivation in your part time team members than your full time team members?

In my experience (we have a 10% equity pool for the team) it is a small part of the puzzle. When it comes to having an engaged, motivated and effective team for me it has all come down to finding the right fit up front, and then me being the leader and visionary on an ongoing basis. In terms of culture that works well for us I've found full-flexibility (with unlimited leave, as long as their job gets done, I'm happy) and a high level of empowerment, trust and autonomy around their role, to be a bigger motivator than equity or money.

I think the remuneration package (including equity) sweetens the deal for sure, and when structured correctly can be a great contributor to motivation, however it seems to be more of a support piece than the main factor.

Hope that helps!

Well, people who do remote jobs, I think one factor that needs to be taken into account is concentration and focus. I always advise staff to look for a very quiet place with good lighting, as in the late afternoon, as the sunset is always inspiring.
The suggestion is to make employees feel important, so important that the work will be done in the best way and that they may not even realize it, because the best songs and the best texts were made that way! Nothing works under pressure.

The most important thing you need to do as a remote team leader is to get agreement on three things. People feel far away from each not because of the number of physical miles, but rather by the amount of time—the delay—it takes to get answer from someone. “Video-first” is an organizational communication strategy that places priority on video conferencing tools, as opposed to audio-only conference calls. Relationships at work are critical to a high performing team. The classic Gallup Q12 survey item, “I have a best friend at work” illustrates the power of workplace friendships on employee engagement. And personal relationships go a long way to building trust and reducing unproductive conflict. Do not forget to recognize team members for their effort and achievements. Any workplace can survive a one or two-week work-from-home experiment; it is not much different than an employee taking a vacation or sick time. But if you expect your team members to work for several months and keep their normal productivity, then you should be prepared to make at least a minimal investment in hardware and software. Great leaders individualize their approach to leadership and take the time to utterly understand what motivates and challenges each team member. Using any of the popular behavioural assessments you can understand who on your team might have an easier or harder time with a work from home situation. Everything DiSC—your team members who have primarily an Influence or Steadiness style are more social than others and may miss the daily interpersonal connections that happen in an office. Your Dominant team members, while needing less social interaction, might struggle with productivity without the structure of the office.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath

Planning your time ,schedule in accordance to your time setting as you want make it easy to get remembered like
1)2:00AM -8:00PM I am in work
2)3:00PM-2:00Am evening I take break or I am online university study
3)3:00PM-5:00PM I would take my time with family
4) 5:00PM I will read some books or study
5)5:30PM I will get to rest
6)6:00AM morning I will start my job again
Time management is all about doing this as your have stated on above not only principles if you do in best way it to profitable