Your DM needs to decide, likely this does not work
Here is the text of the Manifest Echo feature that defines the properties of the echo:
This echo is a magical, translucent,
gray image of you that lasts until it is destroyed,
until you dismiss it as a bonus action, until you manifest
another echo, or until you're incapacitated.
Your echo has AC 14 + your proficiency bonus, 1 hit
point, and immunity to all conditions. If it has to make
a saving throw, it uses your saving throw bonus for the
roll. It is the same size as you, and it occupies its space.
On your turn, you can mentally command the echo to
move up to 30 feet in any direction (no action required).
If your echo is ever more than 30 feet from you at the
end of your turn, it is destroyed.
The Manifest Echo ability does not say the echo has a speed or that is shares your speeds. All it says is that you can command it to move it in any direction. I'm not sure it is intended like that, but as up is a direction, as written this would mean that you could move it upward, too. The language however is different from spiritual weapon, or arcane hand, which says "you can move". Here you can not move the image, you command it to move. It is not clear how it can move without a speed.
See also this question, on wether the echo can move vertically, with split takes on the issue. The majority answer claims yes, the accepted answer, no.
If you decide the image has its own speed of 30 feet you also need to decide if that is a walking speed, or a fly speed. In that case, grappling would affect speed. If you decide you just command the image to move and it will move without a speed, then grappling would not affect its speed, and it could move in any direction.
It also is not clear cut if you can grapple using your echo and hold on to the creature. As that answer explains, Echo Knight is unfortunately written rather sloppy, and your DM must do some work here to define how it will work in their game.
I personally think that Cubic's answer is correct: you can grapple through the echo, but the grapple ends immediately, as the echo is not doing the grappling, you are doing the grappling through the echo. Unless the grappled creature also happens to be in your own reach, the grapple would end immediately, as the rules for the grappled condition say that The condition also ends if an effect removes the grappled creature from the reach of the grappler. You could not move a grappled creature around with the echo, wether upwards or sideways. I would run it that way in my games.
If you allowed both vertical movement and the echo maintaining a grapple, then you could move the target up one round to 30 feet with the grappled creature without it having moved more than 30 feet away from you, and then move it up another 30 feet to 60 feet, at the end of which turn it will be destroyed, and the grappled creature will fall.