How did Christ baptise (wash clean) his little ones?

Did he ever baptise with water from a well or river (like John the Baptise) baptising them in the name of the Father, the Son and the Holy Spirit?

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RE: Did he ever baptise with water from a well or river (like John the Baptise) baptising them in the name of the Father, the Son and the Holy Spirit?

No.

†. John 4:1-3 . .When therefore the Lord knew how the Pharisees had heard that Jesus made and baptized more disciples than John, (Though Jesus himself baptized not, but his disciples) He left Judaea, and departed again into Galilee.

Jesus' mission wasn't to baptize with water like John the Baptist; he was cut out to baptize with different elements altogether.

†. Mtt 3:11 . . I indeed baptize you with water unto repentance: but he that cometh after me is mightier than I, whose shoes I am not worthy to bear: he shall baptize you with the Holy Ghost, and with fire.

In other words, the Bible's Christ doesn't baptize in the name of the Holy Ghost, but rather with the Holy Ghost.

†. 1Cor 12:13 . . For we were all baptized by one Spirit into one body

Water baptisms are applied to the outside of the body; while Christ's baptism is applied to the inside; in point of fact, his baptism is for drinking rather than for bathing.

†. John 4:13-14 . . Jesus answered: Everyone who drinks this water will be thirsty again, but whoever drinks the water I give him will never thirst. Indeed, the water I give him will become in him a spring of water welling up to eternal life.

†. John 7:37-39 . . In the last day, that great day of the feast, Jesus stood and cried, saying, If any man thirst, let him come unto me, and drink. He that believeth on me, as the scripture hath said: out of his belly shall flow rivers of living water. (But this spoke he of The Spirit, which they that believe on him should receive: for the Holy Ghost was not yet given; because that Jesus was not yet glorified).

†. 1Cor 12:13 . . we were all given the one Spirit to drink.

There's no charge for a baptism performed by the Bible's Christ.

†.Rev 22:17 . . And the Spirit and the bride say: Come! And let the one who hears say: Come! And let the one who is thirsty come— let anyone who wishes, take the water of life without cost.

C.L.I.F.F.
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John 4:1-3 . .When therefore the Lord knew how the Pharisees had heard that Jesus made and baptized more disciples than John, (Though Jesus himself baptized not, but his disciples) He left Judaea, and departed again into Galilee.

This is an interesting quote. It tells us that Jesus did not personally baptize people but that he tasked his disciples to baptize his followers.

However, it says "Jesus made and baptized more disciples than John"... This tells us a few things of note.
First, water baptism was not something that came later, it was present from the very beginning of Jesus' ministry. This tends to indicate that it was an essential part of his ministry.
Second, Jesus baptized more people than John the Baptist. John had been prolific often drawing huge crowds of people, but Jesus baptized even more.
Thirdly, even though Jesus himself did not personally baptize these people, it was recognized that his disciples were acting as his agents or representatives and thus when they baptized, it was understood that they did so under the authority and teaching of Jesus, which is why it starts out saying "Jesus made and baptized..." When the disciples baptized, it was the same as if Jesus himself had done it because they were acting on his authority.

as to the mode or method of baptism, it was usually done in a river if such a water source was handy.
The bible doesn't give a lot of detail about the method. The first Christian writing to go into more detail was a work called the Didache, "the teaching of the apostles".
This document is one of the earliest Christian writings and was written during the first century, possibly within the lifetime of the Apostles.

In the didache it says that baptism should be done in "living water" which means moving water like a river, if possible. If there is no source of moving water, then it should be done in a source of cold water, if a source of cold water is not available then it should be done in warm water.
It also says that baptism should be done by immersion if possible, but if this is not practical then it should be done by pouring water over the head of the person three times, invoking the Trinity (Father, Son, and Holy Spirit).

The basic idea presented is that there are preferred forms (dunking in a river) but that ultimately it doesn't matter which form is used, any form will do, the important thing is that the person is baptized with water in the name of the Father, Son, and Holy Spirit.