For reference, I'm working off of Carnie's Syntax (2002). In the book's framework, T → C movement is triggered by a [+Q] feature in C. In the case we have an auxiliary verb, it can raise to T and then to C to make subject-aux inversion, like in Can we eat? In the case we don't have an auxiliary, but rather a null tense head (to mark present or past tense), we get do-support: Carnie says that when there is no other option for supporting inflectional affixes, we insert the dummy verb do into T. This would then move into the C head as in the picture below. However, I'm wondering about sentences such as Who kissed Matt? or Who gave you that present?—since these aren't necessarily echo questions and there are no auxiliaries, why don't we have do-support in these questions? There would be a null past tense head in T, but the way I understand do-support, we must add do to support tense inflection if we have T → C movement in a question; however, something like *Who did kiss Matt? is ungrammatical to me. Could someone help illuminate what the tree and movement operations would be like under this framework?
This problem is addressed in challenge problem set 1 with the example Who ate the pizza?, so you're right in noticing that this is a problem! I am no expert, but I see two options:
The first is to not set [+Q] on the complementizer, so that it does not need a phonological realization. This seems unlikely semantically.
The second would be to say that T-to-C movement of non-auxiliaries is available in English when Spec,TP is empty. Then you are allowed to move the verb from V to T to C, checking [+Q]. In object Wh-questions, Spec,TP is filled with the subject; the movement would be blocked and do-support is required. Recall from the section on do-support that it is always a last resort: it is not used unless absolutely necessary; hence Who did kiss Matt is ungrammatical because you can use kiss to fill the C head and do not need did to do so.
Again, I'm no expert, so there may be problems with this analysis or there may be other solutions still.