Hello,
I try to translate "I send you many kisses" in Esperanto, and I struggle a bit...
Should it be "mi sendas al vi multe da kisoj" or "mi sendas al vi multe da kisojn" or "mi sendas al vi multajn kisojn"? And why?
Thanks in advance

the second option is wrong, because "da" is a preposition, which is never followed by accusativ words.

multajn kisojn is correct, simply

multe da kisoj ankaŭ estas ok, sed for me personally, I don't like this form, because it looks like a break of grammar rules.

In German the translation would be something like: "ich sende dir viel von Küssen"

I would prefer a form like this: mi sendas al vi multon da kisoj, which in German would be crudely: "Ich sende dir ein Vieles von Küssen"

Zamenhof didn't have problems with constructions like: multe da kisoj

As Thyrolf said the second option is wrong.

When it comes to the two other ones... There are three possible ways to use mult/:
 

• Mi sendas al vi multajn kisojn.
  
• Mi sendas al vi multe da kisoj.
  
• Mi sendas al vi multon da kisoj.

multa

The adjective multa has three meanings. It is used for a large unspecified quantity of something, that you cannot express with a number, for instance a tro multa parolado lacigas lin.

It can denote a large unspecified quantity of enumerable things in relation to a smaller quantity, for instance el multaj milonoj fariĝas milionoj.

Without a following noun multa usually refers to people, for instance multaj serĉas favoron de reganto. This is the third meaning.

When it comes to kisses, the second meaning applies here.

multe

Also the adverb multe has three meanings. It is used for a long-lasting or repeated action, for instance ne multe dormu.

It can denote an action concerning a large quantity or is relative intense, for instance kiu multe profitas, ankaŭ perdon ne evitas.

To emphasise comparison. For instance ĝi flugis multe antaŭ ĉiuj aliaj .

When it comes to kisses, the second meaning applies here.

multo

The noun multo has only one meaning, a large quatity. For instance mi neniam prenas kun mi multon da pakaĵoj.

and the difference is?

From the definitions above follows that the adjective describes the amount, the adverb the action and the noun is the amount. Depending on the context these may mean different things, for instance
 

• Ili vendas multajn spicojn kaj oleojn: They sell many kinds of spices and oils.
  
• Ili vendas multe da spicoj kaj oleoj: They sell lots of spices and oils.
  
• Ili vendas multon da spicoj kaj oleoj: They sell great quantities of spices and oils.

But these differences are not always observed by the users. I once searched these three in Tekstaro (a 10+ million word corpus of selected Esperanto texts) and found the following approximate ratio between them: multa to multe to multo is ca. 5 to 10 to 1. So the two first ones are common while the noun is rare.

Having said all this I prefer here mi sendas al vi multajn kisojn perhaps simply because it sounds softer than multe da, which when it comes to kisses gives better vibes