Pushing A Stroller or Wheelchair on Grass or Dirt

Do the same rules that apply to dragging furniture on grass / dirt apply to pushing a stroiller or a wheelchair.

It seems that based on the Mishna that they do not:

מַתְנִי׳ עֲגָלָה שֶׁל קָטָן ...אֵינָהּ נִגְרֶרֶת אֶלָּא עַל גַּבֵּי כֵלִים 

MISHNA: A child’s wagon...may be dragged on the ground on Shabbat only upon cloth, a stone pavement, or the like, as otherwise it would create a furrow when dragged, and one would be liable due to the prohibited labor of plowing.

רַבִּי יְהוּדָה אוֹמֵר כּל הַכֵּלִים אֵין נִגְרָרִין חוּץ מִן הָעֲגָלָה מִפְּנֵי שֶׁהִיא כּוֹבֶשֶׁת: 

Rabbi Yehuda says: For this reason, no vessels may be dragged on the ground on Shabbat except for a wagon, which is permitted because its wheels do not make a furrow in the ground but merely press the earth down. Since no earth is moved from its place, this is not considered digging or plowing on Shabbat.

The difference between dragging furniture and rolling something with wheels is that the firntiture pushes the dirt aside which is similar to plowing, while the wheels of the stroller press down the eart in place. 

This is significant as according to Rashi to definition of plowing is softening the earth. Pushing the dirt aside does that, pressing the earth down, actually does the opposite - it makes the ground harder. 

The concern that the wheels may not turn and the action will be more like dragging than wheels rolling is too far fetched (it is not a Psik Reisha) to enter into the Halachik calculus. 

Rabbi Yaakov Chaim Sofer in his Kaf Hachim (here and continued here rules accordingly)

Rabbi Yosef Tzvi Rimon concurs with this ruling and adds that since the grove is made in an indirect fashion (K'lachar Yad) and it generally ruins the grass (Mikalkel) the entire prohibition is demoted to be of rabbinc origin . 

A few more notes:

Kids can play in a sandbox - the sand is not muktzah since it is set aside to be played with. 

Building with sand is prohibited on shabbat. Building with very dry and loose sand (that will not hold a shape) is permitted on shabbat. Adding water to the sand is also prohibited as a function of kneading. 

Sand on the beach is Muktzah. See here for more on sand.

Walking on the sand on Shabbat is permitted as the moving the inevitably happens in indirect (with your foot) and it is not your intention to do so (Davar Sh'eino Mitkaven) and it is carried out within normal activies (Mitasek).